Counting PARR configurations

This page contains some results with respect to counting the number of PARR's with various properties. This page is generated with the program PARR.cpp. This program went through a number of versions. Only the final version is released and likewise this page presents the results of the last version. The page is devided in the methods that were employed.

First method

The first method I applied was a straight forward counting algorithm. These were generated on August 26, 2015.

All

This table gives the number of all PARR's per number of points (columns) and per number connections (rows), including those with crossing diagonal connections, where all neighbour points are connected.

      1   2   3    4    5    6     7     8     9    10    11    12    13   14   15  16  17 18 19 20
   +-----------------------------------------------------------------------------------------------
  0: 20 135 382  454  194   27     0     0     0     0     0     0     0    0    0   0   0  0  0  0
  1:  0  55 574 1818 2028  641    54     0     0     0     0     0     0    0    0   0   0  0  0  0
  2:  0   0 136 1508 4412 4025   848    36     0     0     0     0     0    0    0   0   0  0  0  0
  3:  0   0  48  771 4286 8095  4422   507     8     0     0     0     0    0    0   0   0  0  0  0
  4:  0   0   0  214 2682 9723 11294  2897   156     0     0     0     0    0    0   0   0  0  0  0
  5:  0   0   0   68 1214 7423 15392  9437   904    22     0     0     0    0    0   0   0  0  0  0
  6:  0   0   0   12  518 4971 16188 17428  4388   141     0     0     0    0    0   0   0  0  0  0
  7:  0   0   0    0   96 2126 12632 23044 11162   811    12     0     0    0    0   0   0  0  0  0
  8:  0   0   0    0   74 1194  8036 22734 20204  3505    56     0     0    0    0   0   0  0  0  0
  9:  0   0   0    0    0  334  4724 19072 26812  9007   366     1     0    0    0   0   0  0  0  0
 10:  0   0   0    0    0  160  2314 13374 27880 17054  1590    12     0    0    0   0   0  0  0  0
 11:  0   0   0    0    0   41  1016  8563 25010 24283  4668    74     0    0    0   0   0  0  0  0
 12:  0   0   0    0    0    0   420  4681 19756 28172 10038   370     0    0    0   0   0  0  0  0
 13:  0   0   0    0    0    0   144  2470 13560 27466 16114  1382     6    0    0   0   0  0  0  0
 14:  0   0   0    0    0    0    36  1106  8712 24137 21590  3582    38    0    0   0   0  0  0  0
 15:  0   0   0    0    0    0     0   408  4926 18595 23766  7147   176    0    0   0   0  0  0  0
 16:  0   0   0    0    0    0     0   182  2532 13211 23460 11304   650    0    0   0   0  0  0  0
 17:  0   0   0    0    0    0     0    31  1258  8532 20486 14783  1670   10    0   0   0  0  0  0
 18:  0   0   0    0    0    0     0     0   450  4904 16172 16816  3590   39    0   0   0  0  0  0
 19:  0   0   0    0    0    0     0     0   208  2835 11934 17014  5648  147    0   0   0  0  0  0
 20:  0   0   0    0    0    0     0     0    34  1218  7664 15252  8058  482    0   0   0  0  0  0
 21:  0   0   0    0    0    0     0     0     0   617  5048 12621  9482 1100    6   0   0  0  0  0
 22:  0   0   0    0    0    0     0     0     0   196  2616  9588 10170 2069   10   0   0  0  0  0
 23:  0   0   0    0    0    0     0     0     0    50  1446  6650  9798 3058   62   0   0  0  0  0
 24:  0   0   0    0    0    0     0     0     0     0   644  4370  8328 4165  190   0   0  0  0  0
 25:  0   0   0    0    0    0     0     0     0     0   220  2564  6928 4780  428   1   0  0  0  0
 26:  0   0   0    0    0    0     0     0     0     0    70  1392  4952 4922  776   0   0  0  0  0
 27:  0   0   0    0    0    0     0     0     0     0     0   719  3516 4747 1228  12   0  0  0  0
 28:  0   0   0    0    0    0     0     0     0     0     0   236  2146 4009 1588  38   0  0  0  0
 29:  0   0   0    0    0    0     0     0     0     0     0    91  1296 3219 1896  96   0  0  0  0
 30:  0   0   0    0    0    0     0     0     0     0     0     2   680 2430 1992 176   0  0  0  0
 31:  0   0   0    0    0    0     0     0     0     0     0     0   280 1532 1872 366   0  0  0  0
 32:  0   0   0    0    0    0     0     0     0     0     0     0   100 1072 1680 407   4  0  0  0
 33:  0   0   0    0    0    0     0     0     0     0     0     0     8  565 1316 634   8  0  0  0
 34:  0   0   0    0    0    0     0     0     0     0     0     0     0  288  988 544  18  0  0  0
 35:  0   0   0    0    0    0     0     0     0     0     0     0     0  110  668 707  72  0  0  0
 36:  0   0   0    0    0    0     0     0     0     0     0     0     0   16  432 506  64  0  0  0
 37:  0   0   0    0    0    0     0     0     0     0     0     0     0    0  240 489 122  0  0  0
 38:  0   0   0    0    0    0     0     0     0     0     0     0     0    0  104 346 170  0  0  0
 39:  0   0   0    0    0    0     0     0     0     0     0     0     0    0   28 232 122  4  0  0
 40:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0 186 186 11  0  0
 41:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0  66 100  0  0  0
 42:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0  38 114 38  0  0
 43:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   1  84 22  0  0
 44:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0  44 20  0  0
 45:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0  28 39  0  0
 46:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0   4 10  0  0
 47:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0   0 32  6  0
 48:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0   0  8  0  0
 49:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0   0  6  0  0
 50:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0   0  0 10  0
 51:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0   0  0  0  0
 52:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0   0  0  4  0
 53:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0   0  0  0  0
 54:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0   0  0  0  0
 55:  0   0   0    0    0    0     0     0     0     0     0     0     0    0    0   0   0  0  0  1

The total number is 1048576. Result match with second method.

And when we add all the PARR's where not all neighbour points are connected, we get the following table:

      1   2    3    4     5      6       7        8         9         10          11          12           13            14             15              16              17               18               19               20
   +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  0: 20 190 1140 4845 15504  38760   77520   125970    167960     184756      167960      125970        77520         38760          15504            4845            1140              190               20                1
  1:  0  55  990 8415 44880 168300  471240  1021020   1750320    2406690     2674100     2406690      1750320       1021020         471240          168300           44880             8415              990               55
  2:  0   0  280 5965 57360 335000 1341200  3925740   8728720   15095080    20592000    22315150     19230640      13114920        6996080         2859500          865200           182680            24040             1485
  3:  0   0   48 2547 45018 415932 2408238  9601956  27861834   60885396   102344814   133931226    137083518     109506852       67623738        31674132        10886238          2590860           381690            26235
  4:  0   0    0  734 25062 368607 3099054 16899015  64231398  177947539   369621054   585501543    713223170     668527629      478620282       257230197       100536114         26995905          4456090           341055
  5:  0   0    0  140 10482 250105 3065592 22912941 114330064  402457075  1035215370  1988926179   2888587682    3182050023     2644858204      1633039515       726606012        220345741         40787074          3478761
  6:  0   0    0   12  3262 133883 2425060 24950216 163810492  734079464  2342542166  5468176315   9483766742   12296315123    11881736752      8438513558      4279025680       1466918426        304766518         28989675
  7:  0   0    0    0   688  56432 1563240 22374290 194119880 1110181508  4403585756 12510297489  25951221398   39660064424    44611405370     36488127904     21109939474       8189137268       1911331234        202927725
  8:  0   0    0    0    74  18165  823658 16747840 193598604 1418772913  7013029490 24295341830  60379583950  108927180170   142825917058    134701349966     89013576656      39115236326      10265684070       1217566350
  9:  0   0    0    0     0   4189  351176 10511589 164171260 1551896171  9595410754 40638203542 121234372742  258604168104   395833916224    431025519558    325724105744     162310156167      47946533470       6358402050
 10:  0   0    0    0     0    611  118430  5515607 118927804 1464810099 11391101012 59179940748 212443274022  536837483473   960775834340   1209611784503   1046647836146     592100187932     197071279086      29248649430
 11:  0   0    0    0     0     41   30392  2395531  73596200 1198379908 11811589342 75624460472 327681845084  983184839633  2061150713728   3004959929815   2981182990868    1916932699344     719655105062     119653565850
 12:  0   0    0    0     0      0    5568   846145  38740386  850903080 10741514616 85275761216 447772649622 1599694651720  3936709486176   6657656249552   7584198626044    5550294130433    2353022543586     438729741450
 13:  0   0    0    0     0      0     648   236494  17199454  523629181  8583588760 85168660376 544667906034 2324864003369  6733027382248  13234787250042  17340263497828   14463138438279    6932720515070    1451182990950
 14:  0   0    0    0     0      0      36    50146   6355346  278150622  6027307636 75500929649 591754716710 3030493210776 10359591658368  23722281051702  35813056335764   34097172767410   18504184590170    4353548972850
 15:  0   0    0    0     0      0       0     7536   1917070  126687531  3712513210 59452098683 575487244068 3553998233112 14391473318092  38491850170466  67095794580296   73043048995136   44943519203774   11899700525790
 16:  0   0    0    0     0      0       0      709    459050   48994223  1998867820 41562099782 501540445324 3757873239123 18101369285748  56721374001795 114426453827272  142702148723690   99709070618798   29749251314475
 17:  0   0    0    0     0      0       0       31     83686   15871668   935716618 25750590332 391769342766 3587296429091 20656661068924  76101935881731 178145189431520  255035810200772  202703280593664   68248282427325
 18:  0   0    0    0     0      0       0        0     10862    4226989   377998486 14097533057 274092169482 3093502189244 21417616888748  93147293884366 253773058130840  418021636291040  378635087038490  144079707346575
 19:  0   0    0    0     0      0       0        0       888     901355   130438840  6790356049 171477957490 2409620281072 20193439435130 104160106199110 331393873701844  629710161308264  651340154624340  280576272201225
 20:  0   0    0    0     0      0       0        0        34     148001    37926678  2860875206  95691812846 1694056522229 17318344072198 106516490377260 397276712265372  873326587014050 1033845512090776  505037289962205
 21:  0   0    0    0     0      0       0        0         0      17579     9119494  1046215372  47465745428 1073403094477 13507061941912  99667809214130 437671460411544 1116622332186482 1516552273888212  841728816603675
 22:  0   0    0    0     0      0       0        0         0       1346     1766118   328768598  20831930212  611680614083  9573067265756  85344154051190 443418004937280 1317627390393492 2058637217347996 1300853625660225
 23:  0   0    0    0     0      0       0        0         0         50      264902    87610880   8041959190  312577046348  6157805166462  66858377433870 413282298431580 1436058765808152 2588656664535700 1866442158555975
 24:  0   0    0    0     0      0       0        0         0          0       28894    19450300   2710335858  142703972680  3588347527370  47885889877435 354378158212320 1446314200685796 3017769339540700 2488589544741300
 25:  0   0    0    0     0      0       0        0         0          0        2040     3510614    789902996   57930984770  1889756701416  31322380973485 279481945210080 1346378868699150 3263314170046684 3085851035479212
 26:  0   0    0    0     0      0       0        0         0          0          70      497337    196652720   20788250513   896640922988  18682400102640 202603898880300 1158451686493320 3274471289679948 3560597348629860
 27:  0   0    0    0     0      0       0        0         0          0           0       52393     41158924    6545707173   381817640764  10140751151600 134879476268160  921033249308080 3049212131234444 3824345300380220
 28:  0   0    0    0     0      0       0        0         0          0           0        3745      7088818    1791879702   145226044756   4996509827492  82355624320848  676281409986090 2634877279399140 3824345300380220
 29:  0   0    0    0     0      0       0        0         0          0           0         151       975256     421473627    49048507672   2227722331682  46043899267464  458232322091710 2112167669212540 3560597348629860
 30:  0   0    0    0     0      0       0        0         0          0           0           2       102608      83894754    14602693352    895338041290  23522582712072  286205804689236 1569874934228724 3085851035479212
 31:  0   0    0    0     0      0       0        0         0          0           0           0         7704      13849020     3797619424    322860520558  10953001101480  164551412655216 1081046085222268 2488589544741300
 32:  0   0    0    0     0      0       0        0         0          0           0           0          364       1843715      852810736    103865270424   4634328777444   86937333092698  689023498241584 1866442158555975
 33:  0   0    0    0     0      0       0        0         0          0           0           0            8        190047      162923060     29602059001   1775198696656   42119368708391  405965178727870 1300853625660225
 34:  0   0    0    0     0      0       0        0         0          0           0           0            0         14218       25959284      7410074958    612905257384   18665248860392  220768867493020  841728816603675
 35:  0   0    0    0     0      0       0        0         0          0           0           0            0           686        3356432      1611681758    189721144286    7543190733932  110604157585986  505037289962205
 36:  0   0    0    0     0      0       0        0         0          0           0           0            0            16         338316       300404875     52314227618    2770055085309   50935737373162  280576272201225
 37:  0   0    0    0     0      0       0        0         0          0           0           0            0             0          24940        47128887     12749599724     920398856273   21505199211786  144079707346575
 38:  0   0    0    0     0      0       0        0         0          0           0           0            0             0           1196         6073972      2719793168     275291750994    8298037784270   68248282427325
 39:  0   0    0    0     0      0       0        0         0          0           0           0            0             0             28          621442       501709474      73664204180    2915478371370   29749251314475
 40:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0           47951        78788656      17502389766     928617758174   11899700525790
 41:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0            2565        10318352       3658307995     266733677798    4353548972850
 42:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0              81         1095410        664866812      68658087014    1451182990950
 43:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               1           90460        103500320      15716215790     438729741450
 44:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0            5444         13528949       3169156890     119653565850
 45:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0             212          1444715        556332326      29248649430
 46:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0               4           121082         83737362       6358402050
 47:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0               0             7472         10591846       1217566350
 48:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0               0              302          1095150        202927725
 49:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0               0                6            88900         28989675
 50:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0               0                0             5314          3478761
 51:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0               0                0              208           341055
 52:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0               0                0                4            26235
 53:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0               0                0                0             1485
 54:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0               0                0                0               55
 55:  0   0    0    0     0      0       0        0         0          0           0           0            0             0              0               0               0                0                0                1

The total number is 84024935266353181. Result match with second method.

No crossings

This table gives the number of all PARR's per number of points (columns) and per number of connections (rows) where all neighbouring points are connected, except for crossing diagonal connections where either one or the other is included.

      1   2   3    4    5    6     7     8     9    10    11    12    13    14    15    16    17    18    19   20
   +-------------------------------------------------------------------------------------------------------------
  0: 20 135 382  454  194   27     0     0     0     0     0     0     0     0     0     0     0     0     0    0
  1:  0  55 574 1818 2028  641    54     0     0     0     0     0     0     0     0     0     0     0     0    0
  2:  0   0 136 1508 4412 4025   848    36     0     0     0     0     0     0     0     0     0     0     0    0
  3:  0   0  48  771 4286 8095  4422   507     8     0     0     0     0     0     0     0     0     0     0    0
  4:  0   0   0  214 2682 9723 11294  2897   156     0     0     0     0     0     0     0     0     0     0    0
  5:  0   0   0   92 1414 7923 15816  9509   904    22     0     0     0     0     0     0     0     0     0    0
  6:  0   0   0    0  466 5237 17312 18456  4476   141     0     0     0     0     0     0     0     0     0    0
  7:  0   0   0    0  208 2872 14412 24948 12206   819    12     0     0     0     0     0     0     0     0    0
  8:  0   0   0    0    6 1116 10100 27592 22900  3981    56     0     0     0     0     0     0     0     0    0
  9:  0   0   0    0    0  510  5960 24510 33216 10991   382     1     0     0     0     0     0     0     0    0
 10:  0   0   0    0    0   48  2742 18304 38784 22334  2166    12     0     0     0     0     0     0     0    0
 11:  0   0   0    0    0    0  1274 12109 37052 35495  7004    74     0     0     0     0     0     0     0    0
 12:  0   0   0    0    0    0   256  6500 30726 46568 15780   682     0     0     0     0     0     0     0    0
 13:  0   0   0    0    0    0     0  3180 22508 49074 30114  2722     6     0     0     0     0     0     0    0
 14:  0   0   0    0    0    0     0  1000 14266 45043 44274  7870    78     0     0     0     0     0     0    0
 15:  0   0   0    0    0    0     0    36  7702 37066 54538 18077   544     0     0     0     0     0     0    0
 16:  0   0   0    0    0    0     0     0  3098 27620 56372 32100  2336     0     0     0     0     0     0    0
 17:  0   0   0    0    0    0     0     0   384 16966 51982 46999  6914    26     0     0     0     0     0    0
 18:  0   0   0    0    0    0     0     0     0  8304 44516 57304 16154   263     0     0     0     0     0    0
 19:  0   0   0    0    0    0     0     0     0  2024 32476 60396 28996  1475     0     0     0     0     0    0
 20:  0   0   0    0    0    0     0     0     0    64 19368 57878 43662  4524     0     0     0     0     0    0
 21:  0   0   0    0    0    0     0     0     0     0  7170 50736 55062 11658    70     0     0     0     0    0
 22:  0   0   0    0    0    0     0     0     0     0   776 37408 59762 22691   514     0     0     0     0    0
 23:  0   0   0    0    0    0     0     0     0     0     0 19176 61252 35378  2318     0     0     0     0    0
 24:  0   0   0    0    0    0     0     0     0     0     0  4384 55994 47120  6872     0     0     0     0    0
 25:  0   0   0    0    0    0     0     0     0     0     0   128 39280 55124 14308    81     0     0     0    0
 26:  0   0   0    0    0    0     0     0     0     0     0     0 15152 61280 24946   704     0     0     0    0
 27:  0   0   0    0    0    0     0     0     0     0     0     0  1600 58592 35548  3140     0     0     0    0
 28:  0   0   0    0    0    0     0     0     0     0     0     0     0 34888 46874  6904     0     0     0    0
 29:  0   0   0    0    0    0     0     0     0     0     0     0     0  8464 58520 14668    32     0     0    0
 30:  0   0   0    0    0    0     0     0     0     0     0     0     0   256 53792 24248   816     0     0    0
 31:  0   0   0    0    0    0     0     0     0     0     0     0     0     0 24480 35008  2440     0     0    0
 32:  0   0   0    0    0    0     0     0     0     0     0     0     0     0  2688 51968  7688     0     0    0
 33:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0 41344 11488     0     0    0
 34:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0 11264 26240   448     0    0
 35:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0   256 39424  2752     0    0
 36:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0 23552  3968     0    0
 37:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0  2048 17664     0    0
 38:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0 24576     0    0
 39:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0  6144  1536    0
 40:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0     0 10240    0
 41:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0     0  8192    0
 42:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0     0     0    0
 43:  0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0     0     0 4096

The total number is 2976416. Result match with second method.

And when we add all the PARR's where not all neighbour points are connected, we get the following table:

      1   2    3    4     5      6       7        8         9         10         11          12           13            14            15             16             17              18              19              20
   +------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  0: 20 190 1140 4845 15504  38760   77520   125970    167960     184756     167960      125970        77520         38760         15504           4845           1140             190              20               1
  1:  0  55  990 8415 44880 168300  471240  1021020   1750320    2406690    2674100     2406690      1750320       1021020        471240         168300          44880            8415             990              55
  2:  0   0  280 5953 57168 333560 1334480  3903900   8676304   14998984   20454720    22160710     19093360      13018824       6943664        2837660         858480          181240           23848            1473
  3:  0   0   48 2499 44090 407344 2358566  9402848  27277978   59592104  100138038   130998582    134035902     107036384      66076738       30939944       10630766         2529352          372530           25599
  4:  0   0    0  662 23202 345994 2930828 16048887  61144762  169616964  352548108   558596841    680444534     637692858     456417172      245212289       95801838        25714072         4242708          324585
  5:  0   0    0   92  8482 215352 2718578 20649748 104014972  368245579  950528394  1830055071   2661026546    2933100718    2438377662     1505399642      669616768       202976623        37552438         3201015
  6:  0   0    0    0  2022  99242 1936902 20734868 139339902  633449990 2039909196  4789582049   8337635074   10834859618   10483052958     7449627194     3778002282      1294852646       268886976        25559705
  7:  0   0    0    0   256  33037 1067670 16539886 150285662  884510828 3573839672 10276764805  21488904622   33012027331   37255819872    30531027950    17680589980      6860592156      1600843434       169855175
  8:  0   0    0    0     6   7422  450356 10538333 132091058 1017969522 5198595826 18403695102  46407535828   84542552624  111570482930   105657091470    69989359800     30791612506      8083262940       958311765
  9:  0   0    0    0     0    966  140886  5335144  95024964  974156277 6348837386 27851450380  85098239898  184508798211  285559926438   313232723698   237803377624    118809812339     35136422854      4659693035
 10:  0   0    0    0     0     48   30744  2111473  55821904  778010306 6553787514 35922652361 133766279624  346675410928  631334770388   804282063988   701326336748    398626360220    133004904504     19755322853
 11:  0   0    0    0     0      0    4146   633855  26541030  518179129 5736836700 39701750317 181458308622  565016264366 1215462890024  1803819047542  1811045635858   1173423901704    442453213418     73703836155
 12:  0   0    0    0     0      0     256   137089  10048042  286380318 4258721692 37705341693 213373541238  803146705714 2050176872498  3556832672930  4123132844562   3052132079508   1302920964436    243777975309
 13:  0   0    0    0     0      0       0    19491   2950672  130085566 2673505660 30789195947 218041931406  999425503010 3043486167124  6197403762944  8320777259496   7054684765347   3416430357150    719140541907
 14:  0   0    0    0     0      0       0     1508    644938   47841456 1410996004 21583469299 193794588960 1091243471866 3989093863844  9578318814452 14947893870014  14556290075634   8014941936176   1901463081405
 15:  0   0    0    0     0      0       0       36     98018   13928367  620241140 12940770611 149699224782 1046494728692 4625931104632 13167671270428 23982447439656  26908536186780  16887959780706   4524496838179
 16:  0   0    0    0     0      0       0        0      9130    3103544  224008110  6596019078 100266751790  881238672870 4751168505236 16131771764353 34447666563324  44691094592730  32058973364292   9720614193955
 17:  0   0    0    0     0      0       0        0       384     502044   65188866  2833043880  57996839790  650673450056 4322297807060 17630507220160 44372533072096  66830077375725  54964534145850  18906830723277
 18:  0   0    0    0     0      0       0        0         0      53888   14861662  1012807777  28799598496  420118442486 3479915987072 17194034371436 51309789509304  90117061071840  85269507586200  33363360520371
 19:  0   0    0    0     0      0       0        0         0       3208    2544546   296285381  12177011852  236237311282 2474967960554 14955947833508 53282747987826 109685135775480 119861219359350  53500976743725
 20:  0   0    0    0     0      0       0        0         0         64     305680    69254604   4335660836  115036526628 1550510133768 11588676385157 49679222894354 120555535914480 152801546391900  78058303626555
 21:  0   0    0    0     0      0       0        0         0          0      22746    12495720   1280695800   48144644064  852195530534  7982665161322 41552500807056 119645361558525 176740898386050 103702636455525
 22:  0   0    0    0     0      0       0        0         0          0        776     1650932    307507414   17145626924  408728214382  4873980198250 31131943199814 107157391741290 185487103283040 125502767053515
 23:  0   0    0    0     0      0       0        0         0          0          0      146200     58321516    5127848793  169872936152  2627453128534 20847481311196  86513667497940 176553288323550 138364589845845
 24:  0   0    0    0     0      0       0        0         0          0          0        7296      8375006    1265467770   60628972618  1244187375332 12441049622064  62857724203590 152281509017940 138917164933575
 25:  0   0    0    0     0      0       0        0         0          0          0         128       851152     251552537   18365651006   514164264098  6591016138776  41006483067105 118862973301410 126919671598521
 26:  0   0    0    0     0      0       0        0         0          0          0           0        54128      38917962    4649499756   183895814708  3084789221804  23948346073260  83805133030920 105402262904055
 27:  0   0    0    0     0      0       0        0         0          0          0           0         1600       4449200     963566300    56321309950  1267699308366  12473581175800  53243760917678  79439115953833
 28:  0   0    0    0     0      0       0        0         0          0          0           0            0        344952     158823762    14568460600   453923099310   5767040870052  30388461553980  54223568712079
 29:  0   0    0    0     0      0       0        0         0          0          0           0            0         15376      19960392     3125149692   140249689864   2352965524657  15521347338538  33433386823185
 30:  0   0    0    0     0      0       0        0         0          0          0           0            0           256       1788640      542304184    36930162002    841023595198   7061158017968  18561289434015
 31:  0   0    0    0     0      0       0        0         0          0          0           0            0             0        101152       73477216     8154589728    260948115316   2844528116886   9241603253505
 32:  0   0    0    0     0      0       0        0         0          0          0           0            0             0          2688        7366720     1477713384     69470761096   1007362453624   4106643801600
 33:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0         499584      213279456     15631885856    310790135040   1618972188440
 34:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0          19200       23467648      2913937824     82577842944    562099627760
 35:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0            256        1836544       437702080     18617262144    170307464688
 36:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0              0          90112        50877824      3491630848     44511668480
 37:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0              0           2048         4291328       530036736      9886513664
 38:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0              0              0          233472        62566400      1829436672
 39:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0              0              0            6144         5387776       274394880
 40:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0              0              0               0          301056        32047104
 41:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0              0              0               0            8192         2734080
 42:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0              0              0               0               0          151552
 43:  0   0    0    0     0      0       0        0         0          0          0           0            0             0             0              0              0               0               0            4096

The total number is 4147603839035069. Result match with second method.

Second method

The second method I applied involved slicing. These were generated on September 3, 2015.

Connected

This table gives the number of all PARR's per number of points (columns) and per number connections (rows), including those with crossing diagonal connections, where all neighbour points are connected and where all points are connected through one or more connections.

      1  2   3   4   5    6    7    8     9    10    11    12   13   14   15  16  17 18 19 20
   +-----------------------------------------------------------------------------------------
  0: 20  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0  0  0  0
  1:  0 55   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0  0  0  0
  2:  0  0 136   0   0    0    0    0     0     0     0     0    0    0    0   0   0  0  0  0
  3:  0  0  48 347   0    0    0    0     0     0     0     0    0    0    0   0   0  0  0  0
  4:  0  0   0 214 746    0    0    0     0     0     0     0    0    0    0   0   0  0  0  0
  5:  0  0   0  68 766 1301    0    0     0     0     0     0    0    0    0   0   0  0  0  0
  6:  0  0   0  12 418 2027 1766    0     0     0     0     0    0    0    0   0   0  0  0  0
  7:  0  0   0   0  96 1556 3882 1662     0     0     0     0    0    0    0   0   0  0  0  0
  8:  0  0   0   0  74  750 4138 5210  1040     0     0     0    0    0    0   0   0  0  0  0
  9:  0  0   0   0   0  334 2892 7674  4544   297     0     0    0    0    0   0   0  0  0  0
 10:  0  0   0   0   0  160 1646 7226  9254  2332     6     0    0    0    0   0   0  0  0  0
 11:  0  0   0   0   0   41  830 5225 11610  6859   506     0    0    0    0   0   0  0  0  0
 12:  0  0   0   0   0    0  420 3394 10958 11670  2520     8    0    0    0   0   0  0  0  0
 13:  0  0   0   0   0    0  144 1982  8780 13904  6408   304    0    0    0   0   0  0  0  0
 14:  0  0   0   0   0    0   36  986  6326 14213 10528  1470    2    0    0   0   0  0  0  0
 15:  0  0   0   0   0    0    0  408  3974 12599 13408  3647   80    0    0   0   0  0  0  0
 16:  0  0   0   0   0    0    0  182  2124  9787 14978  6440  408    0    0   0   0  0  0  0
 17:  0  0   0   0   0    0    0   31  1174  6764 14204  9607 1060    8    0   0   0  0  0  0
 18:  0  0   0   0   0    0    0    0   450  4250 12340 11574 2484   35    0   0   0  0  0  0
 19:  0  0   0   0   0    0    0    0   208  2479  9500 12562 4204  135    0   0   0  0  0  0
 20:  0  0   0   0   0    0    0    0    34  1158  6636 11850 6124  404    0   0   0  0  0  0
 21:  0  0   0   0   0    0    0    0     0   617  4422 10381 7546  948    6   0   0  0  0  0
 22:  0  0   0   0   0    0    0    0     0   196  2428  8106 8314 1773   10   0   0  0  0  0
 23:  0  0   0   0   0    0    0    0     0    50  1382  5950 8316 2664   62   0   0  0  0  0
 24:  0  0   0   0   0    0    0    0     0     0   644  3970 7174 3669  186   0   0  0  0  0
 25:  0  0   0   0   0    0    0    0     0     0   220  2448 6178 4212  414   1   0  0  0  0
 26:  0  0   0   0   0    0    0    0     0     0    70  1340 4520 4466  748   0   0  0  0  0
 27:  0  0   0   0   0    0    0    0     0     0     0   719 3280 4255 1158  12   0  0  0  0
 28:  0  0   0   0   0    0    0    0     0     0     0   236 2074 3757 1492  38   0  0  0  0
 29:  0  0   0   0   0    0    0    0     0     0     0    91 1264 2981 1796  96   0  0  0  0
 30:  0  0   0   0   0    0    0    0     0     0     0     2  680 2300 1876 176   0  0  0  0
 31:  0  0   0   0   0    0    0    0     0     0     0     0  280 1492 1820 366   0  0  0  0
 32:  0  0   0   0   0    0    0    0     0     0     0     0  100 1056 1564 396   4  0  0  0
 33:  0  0   0   0   0    0    0    0     0     0     0     0    8  565 1288 622   8  0  0  0
 34:  0  0   0   0   0    0    0    0     0     0     0     0    0  288  952 536  18  0  0  0
 35:  0  0   0   0   0    0    0    0     0     0     0     0    0  110  664 683  72  0  0  0
 36:  0  0   0   0   0    0    0    0     0     0     0     0    0   16  432 506  64  0  0  0
 37:  0  0   0   0   0    0    0    0     0     0     0     0    0    0  240 469 122  0  0  0
 38:  0  0   0   0   0    0    0    0     0     0     0     0    0    0  104 346 170  0  0  0
 39:  0  0   0   0   0    0    0    0     0     0     0     0    0    0   28 232 122  4  0  0
 40:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0 186 182 11  0  0
 41:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0  66 100  0  0  0
 42:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0  38 114 38  0  0
 43:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   1  84 22  0  0
 44:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0  44 20  0  0
 45:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0  28 39  0  0
 46:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   4 10  0  0
 47:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0 32  6  0
 48:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0  8  0  0
 49:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0  6  0  0
 50:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0  0 10  0
 51:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0  0  0  0
 52:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0  0  4  0
 53:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0  0  0  0
 54:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0  0  0  0
 55:  0  0   0   0   0    0    0    0     0     0     0     0    0    0    0   0   0  0  0  1

The total number is 517646.

And when we add all the PARR's where not all neighbour points are connected, we get the following table:

      1  2   3    4     5     6      7       8        9        10         11          12           13            14            15             16              17              18               19               20
   +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  0: 20  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0               0                0                0
  1:  0 55   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0               0                0                0
  2:  0  0 280    0     0     0      0       0        0         0          0           0            0             0             0              0               0               0                0                0
  3:  0  0  48 1731     0     0      0       0        0         0          0           0            0             0             0              0               0               0                0                0
  4:  0  0   0  734 11446     0      0       0        0         0          0           0            0             0             0              0               0               0                0                0
  5:  0  0   0  140  8242 76707      0       0        0         0          0           0            0             0             0              0               0               0                0                0
  6:  0  0   0   12  3070 80819 504804       0        0         0          0           0            0             0             0              0               0               0                0                0
  7:  0  0   0    0   688 45684 716324 3195147        0         0          0           0            0             0             0              0               0               0                0                0
  8:  0  0   0    0    74 17055 550978 5790807 19158284         0          0           0            0             0             0              0               0               0                0                0
  9:  0  0   0    0     0  4189 289950 5696530 42767084 107378885          0           0            0             0             0              0               0               0                0                0
 10:  0  0   0    0     0   611 109876 3891534 51729468 287551087  555301346           0            0             0             0              0               0               0                0                0
 11:  0  0   0    0     0    41  29818 1985790 43755656 416069806 1748407184  2613812932            0             0             0              0               0               0                0                0
 12:  0  0   0    0     0     0   5568  772644 28168260 422319655 2965419576  9523974288  11029666090             0             0              0               0               0                0                0
 13:  0  0   0    0     0     0    648  228070 14266308 329634951 3532290098 18639311753  45911541360   40985207931             0              0               0               0                0                0
 14:  0  0   0    0     0     0     36   49678  5740722 206028550 3254583758 25625328279 102375981388  192806501075  131159801488              0               0               0                0                0
 15:  0  0   0    0     0     0      0    7536  1825822 104979305 2426564422 27347371125 160327729446  484721279080  690975003006   350998859857               0               0                0                0
 16:  0  0   0    0     0     0      0     709   450542  43832087 1497562894 23775221256 195341384496  855466493059 1941122499558  2054620219255    753346849574               0                0                0
 17:  0  0   0    0     0     0      0      31    83314  14938910  773140112 17277904302 194760035294 1176296434129 3825868019592  6400503325181   4866878070010   1214554530192                0                0
 18:  0  0   0    0     0     0      0       0    10862   4106905  334844802 10642716090 163406398494 1327793401733 5880497700466 13979209159887  16701417009116   8609029126058    1305156949328                0
 19:  0  0   0    0     0     0      0       0      888    891587  121289710  5593897513 117248918534 1267335414339 7437052451942 23822866705673  40151876339814  32358563873032   10097221259440     698512774464
 20:  0  0   0    0     0     0      0       0       34    147627   36434018  2513130916  72606796354 1041109997234 7981612351076 33460381820024  75336342005074  85136797208946   41356216220186    5868462472880
 21:  0  0   0    0     0     0      0       0        0     17579    8943164   962847311  38978341484  744069688302 7408140781172 39989071378555 116635341985252 174822061994870  118468106595874   26061463099727
 22:  0  0   0    0     0     0      0       0        0      1346    1752658   312692893  18160349966  465542465815 6017987124318 41483395716150 153955712064046 296437584008148  264812843079836   80870614707470
 23:  0  0   0    0     0     0      0       0        0        50     264402    85210882   7330703214  255789002618 4311018816444 37838821313938 176885546796808 429179612440308  489021141345216  195748679512336
 24:  0  0   0    0     0     0      0       0        0         0      28894    19189586   2552848914  123503606318 2735487261336 30606403180007 179331460635980 541977013931315  771825375035048  391483492205329
 25:  0  0   0    0     0     0      0       0        0         0       2040     3492254    761578360   52331415274 1540918163806 22073770386456 161918099932176 605560197119250 1064146343529574  669544954835550
 26:  0  0   0    0     0     0      0       0        0         0         70      496707    192654764   19395737597  770905352164 14242030490505 131015069002248 604547412841322 1300705353147134 1001290972829373
 27:  0  0   0    0     0     0      0       0        0         0          0       52393     40739040    6254909513  342145349620  8234233080498  95395832112158 542958721981966 1423996747305434 1329287985542168
 28:  0  0   0    0     0     0      0       0        0         0          0        3745      7058858    1741933992  134362789376  4267437576466  62665906331156 440782276285600 1406527256592072 1583294011422524
 29:  0  0   0    0     0     0      0       0        0         0          0         151       974048     414621087   46495780208  1980715711602  37187716361788 324482615823066 1259920115421014 1704845178132546
 30:  0  0   0    0     0     0      0       0        0         0          0           2       102592      83175384   14095446848   821698268353  19941403369542 217041818464282 1027250384830932 1668677645624225
 31:  0  0   0    0     0     0      0       0        0         0          0           0         7704      13795078    3714058888   303708109269   9656513978090 132050102116468  764233583528826 1490538757372983
 32:  0  0   0    0     0     0      0       0        0         0          0           0          364       1841167     841709318    99567881414   4216314459718  73093950136093  519601047872342 1218472427648540
 33:  0  0   0    0     0     0      0       0        0         0          0           0            8        189991     161780750    28782213742   1655955605290  36792217089195  323111914923942  913304357535542
 34:  0  0   0    0     0     0      0       0        0         0          0           0            0         14218      25873928     7279660278    583058490390  16820626392460  183793400146488  628417757215166
 35:  0  0   0    0     0     0      0       0        0         0          0           0            0           686       3352316     1594847510    183234461388   6971291954758   95585020006276  397143897342334
 36:  0  0   0    0     0     0      0       0        0         0          0           0            0            16        338220      298710499     51106578290   2612268660949   45400996299934  230518219558580
 37:  0  0   0    0     0     0      0       0        0         0          0           0            0             0         24940       47004115     12560393668    881951231217   19662101167804  122825043274899
 38:  0  0   0    0     0     0      0       0        0         0          0           0            0             0          1196        6067992      2695440556    267095495772    7746093855606   60012364530332
 39:  0  0   0    0     0     0      0       0        0         0          0           0            0             0            28         621302       499220718     72153719108    2767716620840   26846843057720
 40:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0          47951        78596776     17265431142     893512111412   10973319351896
 41:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0           2565        10308092      3627305741     259398882748    4087096978457
 42:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0             81         1095086       661578164      67325806784    1382541756860
 43:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              1           90456       103228880      15508944546     423010745541
 44:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0            5444        13512617       3142078344     116483335979
 45:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0             212         1444079        553441884      28692126881
 46:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               4          121070         83495174       6274642458
 47:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0            7472         10576902       1206972750
 48:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0             302          1094546        201832489
 49:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0               6            88888         28900773
 50:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0               0             5314          3473447
 51:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0               0              208           340847
 52:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0               0                4            26231
 53:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0               0                0             1485
 54:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0               0                0               55
 55:  0  0   0    0     0     0      0       0        0         0          0           0            0             0             0              0               0               0                0                1

The total number is 31111752828628268.

Connected and no crossings

This table gives the number of all PARR's per number of points (columns) and per number of connections (rows) where all neighbouring points are connected, except for crossing diagonal connections where either one or the other is included. Futhermore, all points are connected through one or more connections.

      1  2   3   4   5    6    7    8     9    10    11    12    13    14    15    16    17    18    19   20
   +--------------------------------------------------------------------------------------------------------
  0: 20  0   0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0    0
  1:  0 55   0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0    0
  2:  0  0 136   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0    0
  3:  0  0  48 347   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0    0
  4:  0  0   0 214 746    0    0    0     0     0     0     0     0     0     0     0     0     0     0    0
  5:  0  0   0  92 766 1301    0    0     0     0     0     0     0     0     0     0     0     0     0    0
  6:  0  0   0   0 466 2027 1766    0     0     0     0     0     0     0     0     0     0     0     0    0
  7:  0  0   0   0 208 1720 3882 1662     0     0     0     0     0     0     0     0     0     0     0    0
  8:  0  0   0   0   6 1092 4514 5210  1040     0     0     0     0     0     0     0     0     0     0    0
  9:  0  0   0   0   0  510 3856 8294  4544   297     0     0     0     0     0     0     0     0     0    0
 10:  0  0   0   0   0   48 2566 9148  9878  2332     6     0     0     0     0     0     0     0     0    0
 11:  0  0   0   0   0    0 1274 8241 14474  7187   506     0     0     0     0     0     0     0     0    0
 12:  0  0   0   0   0    0  256 5756 16786 14238  2616     8     0     0     0     0     0     0     0    0
 13:  0  0   0   0   0    0    0 3180 15800 21526  7624   304     0     0     0     0     0     0     0    0
 14:  0  0   0   0   0    0    0 1000 12074 25933 16208  1750     2     0     0     0     0     0     0    0
 15:  0  0   0   0   0    0    0   36  7646 26506 25952  5559    80     0     0     0     0     0     0    0
 16:  0  0   0   0   0    0    0    0  3098 22952 33436 13430   600     0     0     0     0     0     0    0
 17:  0  0   0   0   0    0    0    0   384 16486 37506 24155  2664     8     0     0     0     0     0    0
 18:  0  0   0   0   0    0    0    0     0  8304 36912 35324  7306    75     0     0     0     0     0    0
 19:  0  0   0   0   0    0    0    0     0  2024 30684 43324 16188   659     0     0     0     0     0    0
 20:  0  0   0   0   0    0    0    0     0    64 19336 47846 28362  2368     0     0     0     0     0    0
 21:  0  0   0   0   0    0    0    0     0     0  7170 46800 39522  6890    38     0     0     0     0    0
 22:  0  0   0   0   0    0    0    0     0     0   776 37152 48834 15331   370     0     0     0     0    0
 23:  0  0   0   0   0    0    0    0     0     0     0 19176 55300 26578  1470     0     0     0     0    0
 24:  0  0   0   0   0    0    0    0     0     0     0  4384 55098 38112  4616     0     0     0     0    0
 25:  0  0   0   0   0    0    0    0     0     0     0   128 39280 48532 11732    81     0     0     0    0
 26:  0  0   0   0   0    0    0    0     0     0     0     0 15152 59360 20498   512     0     0     0    0
 27:  0  0   0   0   0    0    0    0     0     0     0     0  1600 58592 30812  2628     0     0     0    0
 28:  0  0   0   0   0    0    0    0     0     0     0     0     0 34888 44058  6136     0     0     0    0
 29:  0  0   0   0   0    0    0    0     0     0     0     0     0  8464 58520 13132    32     0     0    0
 30:  0  0   0   0   0    0    0    0     0     0     0     0     0   256 53792 21688   816     0     0    0
 31:  0  0   0   0   0    0    0    0     0     0     0     0     0     0 24480 35008  2440     0     0    0
 32:  0  0   0   0   0    0    0    0     0     0     0     0     0     0  2688 51968  6664     0     0    0
 33:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0 41344 11488     0     0    0
 34:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0 11264 26240   448     0    0
 35:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0   256 39424  2752     0    0
 36:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0     0 23552  3968     0    0
 37:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0     0  2048 17664     0    0
 38:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0     0     0 24576     0    0
 39:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0     0     0  6144  1536    0
 40:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0 10240    0
 41:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0  8192    0
 42:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0    0
 43:  0  0   0   0   0    0    0    0     0     0     0     0     0     0     0     0     0     0     0 4096

The total number is 2041600.

And when we add all the PARR's where not all neighbour points are connected, we get the following table:

      1  2   3    4     5     6      7       8        9        10         11         12          13           14            15            16             17             18             19             20
   +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  0: 20  0   0    0     0     0      0       0        0         0          0          0           0            0             0             0              0              0              0              0
  1:  0 55   0    0     0     0      0       0        0         0          0          0           0            0             0             0              0              0              0              0
  2:  0  0 280    0     0     0      0       0        0         0          0          0           0            0             0             0              0              0              0              0
  3:  0  0  48 1683     0     0      0       0        0         0          0          0           0            0             0             0              0              0              0              0
  4:  0  0   0  662 10738     0      0       0        0         0          0          0           0            0             0             0              0              0              0              0
  5:  0  0   0   92  7010 68926      0       0        0         0          0          0           0            0             0             0              0              0              0              0
  6:  0  0   0    0  2022 65050 431404       0        0         0          0          0           0            0             0             0              0              0              0              0
  7:  0  0   0    0   256 29197 544292 2580344        0         0          0          0           0            0             0             0              0              0              0              0
  8:  0  0   0    0     6  7332 337222 4139440 14524166         0          0          0           0            0             0             0              0              0              0              0
  9:  0  0   0    0     0   966 127194 3315034 28616570  75877913          0          0           0            0             0             0              0              0              0              0
 10:  0  0   0    0     0    48  30072 1684167 28406978 179004422  362923658          0           0            0             0             0              0              0              0              0
 11:  0  0   0    0     0     0   4146  578633 18306260 213921999 1005307698 1566585275           0            0             0             0              0              0              0              0
 12:  0  0   0    0     0     0    256  133761  8305430 168294750 1415455440 5016490573  6005570758            0             0             0              0              0              0              0
 13:  0  0   0    0     0     0      0   19491  2710308  95444340 1323805934 8182992354 21947742742  20061354848             0             0              0              0              0              0
 14:  0  0   0    0     0     0      0    1508   626842  40389214  906082586 8921137517 40917643730  82772878346   57030968412             0              0              0              0              0
 15:  0  0   0    0     0     0      0      36    97586  12820905  473302748 7191222660 51205021688 174370881513  263212182018  133763320998              0              0              0              0
 16:  0  0   0    0     0     0      0       0     9130   3002496  191619924 4490951190 47728173620 247332772794  620493546738  685041720246   247766521704              0              0              0
 17:  0  0   0    0     0     0      0       0      384    497820   60031480 2219773022 34834629606 262708791837  987009830438 1792085625434  1398169877744   338579200929              0              0
 18:  0  0   0    0     0     0      0       0        0     53888   14316774  873945735 20416889346 220202310548 1180279980544 3167909012572  4029502329410  2092068964952   301925803700              0
 19:  0  0   0    0     0     0      0       0        0      3208    2512466  272773567  9718223116 149788282522 1120210684494 4221687166342  7854194394280  6599149829668  2030600018182   130899473007
 20:  0  0   0    0     0     0      0       0        0        64     305040   66464852  3763109448  83892027754  869548468750 4484302625584 11564281866224 14085211317426  6967449502708   952573320322
 21:  0  0   0    0     0     0      0       0        0         0      22746   12290270  1178184710  38908277864  561350300724 3918248006256 13614296053560 22740439049917 16177951416588  3533500266708
 22:  0  0   0    0     0     0      0       0        0         0        776    1643948   294091866  14931250214  303935421286 2868605553402 13241728096504 29422991971122 28436948387836  8867062304725
 23:  0  0   0    0     0     0      0       0        0         0          0     146200    57144348   4710035985  138370630060 1777943698372 10852213039910 31554056914994 40120332283598 16849342871429
 24:  0  0   0    0     0     0      0       0        0         0          0       7296     8316638   1205898948   52852255872  937386226700  7582575807042 28634547003882 47024452839752 25720740061986
 25:  0  0   0    0     0     0      0       0        0         0          0        128      850128    245544377   16824505608  420653098790  4545977700024 22272018126075 46782629618830 32666154511061
 26:  0  0   0    0     0     0      0       0        0         0          0          0       54128     38538234    4412297312  160178464296  2344429959648 14961186391334 40046736998704 35287151298718
 27:  0  0   0    0     0     0      0       0        0         0          0          0        1600      4438000     936612748   51412700530  1039329206090  8713391563548 29747553354234 32887627744327
 28:  0  0   0    0     0     0      0       0        0         0          0          0           0       344952     156745922   13762219358   394663928298  4403826535784 19267704818226 26687760312193
 29:  0  0   0    0     0     0      0       0        0         0          0          0           0        15376      19868136    3024301492   127536222992  1928455489853 10903965804954 18960808049481
 30:  0  0   0    0     0     0      0       0        0         0          0          0           0          256       1787104     533311640    34728646018   728782234320  5390015803194 11827473101223
 31:  0  0   0    0     0     0      0       0        0         0          0          0           0            0        101152      72970688     7857361280   236135861028  2321697470374  6481734421075
 32:  0  0   0    0     0     0      0       0        0         0          0          0           0            0          2688       7353280     1448039144    64979645520   867499916240  3116680764056
 33:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0        499584      211275232    14985468608   279255670976  1310779739816
 34:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0         19200       23390848     2843071776    76700349440   479726348528
 35:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0           256        1835520      432177088    17735861568   151673035952
 36:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0             0          90112       50607488     3389431808    41011678464
 37:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0             0           2048        4285184      521438720     9355155712
 38:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0             0              0         233472       62099456     1766759680
 39:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0             0              0           6144        5375488      269003008
 40:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0             0              0              0         301056       31746048
 41:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0             0              0              0           8192        2725888
 42:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0             0              0              0              0         151552
 43:  0  0   0    0     0     0      0       0        0         0          0          0           0            0             0             0              0              0              0           4096

The total number is 822802321541410.

Unique configurations

The connected configurations still contain many configurations that only differ with respect to orientation and translation. I worked on a method for finding these. Below the tables for the various kinds of unique connected PARR configurations are given. (These were generated on
September 26, 2015.)

All the unique PARR configurations (on 9 or fewer points) without restrictions are:

     1 2 3  4   5    6     7      8       9
   +---------------------------------------
  0: 1 0 0  0   0    0     0      0       0
  1: 0 2 0  0   0    0     0      0       0
  2: 0 0 6  0   0    0     0      0       0
  3: 0 0 1 40   0    0     0      0       0
  4: 0 0 0 15 302    0     0      0       0
  5: 0 0 0  4 175 2839     0      0       0
  6: 0 0 0  1  60 2427 26621      0       0
  7: 0 0 0  0  13 1192 31301 239977       0
  8: 0 0 0  0   2  406 20456 372703 1978587
  9: 0 0 0  0   0   99  9401 315932 3931302
 10: 0 0 0  0   0   16  3214 188166 4218273
 11: 0 0 0  0   0    2   802  85034 3166605
 12: 0 0 0  0   0    0   145  29745 1815558
 13: 0 0 0  0   0    0    17   8044  823089
 14: 0 0 0  0   0    0     2   1634  298662
 15: 0 0 0  0   0    0     0    243   86328
 16: 0 0 0  0   0    0     0     24   19574
 17: 0 0 0  0   0    0     0      2    3355
 18: 0 0 0  0   0    0     0      0     418
 19: 0 0 0  0   0    0     0      0      33
 20: 0 0 0  0   0    0     0      0       2

The total number is 17682852.

The unique PARR configurations with no crossing diagonals are:

     1 2 3  4   5    6     7      8       9
   +---------------------------------------
  0: 1 0 0  0   0    0     0      0       0
  1: 0 2 0  0   0    0     0      0       0
  2: 0 0 6  0   0    0     0      0       0
  3: 0 0 1 39   0    0     0      0       0
  4: 0 0 0 13 291    0     0      0       0
  5: 0 0 0  3 155 2655     0      0       0
  6: 0 0 0  0  44 2056 24053      0       0
  7: 0 0 0  0   6  821 25550 206825       0
  8: 0 0 0  0   1  187 13659 288919 1604204
  9: 0 0 0  0   0   28  4519 202497 2856320
 10: 0 0 0  0   0    2   977  90390 2555288
 11: 0 0 0  0   0    0   127  27630 1479539
 12: 0 0 0  0   0    0    10   5763  602717
 13: 0 0 0  0   0    0     0    800  176797
 14: 0 0 0  0   0    0     0     63   37029
 15: 0 0 0  0   0    0     0      4    5270
 16: 0 0 0  0   0    0     0      0     472
 17: 0 0 0  0   0    0     0      0      20

The total number is 10215753.

The unique PARR configurations where all neighbour points have a connection, are:

     1 2 3  4  5   6   7    8    9   10   11   12   13   14  15  16 17 18 19 20
   +---------------------------------------------------------------------------
  0: 1 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  0  0  0
  1: 0 2 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  0  0  0
  2: 0 0 4  0  0   0   0    0    0    0    0    0    0    0   0   0  0  0  0  0
  3: 0 0 1 14  0   0   0    0    0    0    0    0    0    0   0   0  0  0  0  0
  4: 0 0 0  5 50   0   0    0    0    0    0    0    0    0   0   0  0  0  0  0
  5: 0 0 0  2 26 148   0    0    0    0    0    0    0    0   0   0  0  0  0  0
  6: 0 0 0  1 12 156 323    0    0    0    0    0    0    0   0   0  0  0  0  0
  7: 0 0 0  0  3  87 548  374    0    0    0    0    0    0   0   0  0  0  0  0
  8: 0 0 0  0  2  29 446 1048  269    0    0    0    0    0   0   0  0  0  0  0
  9: 0 0 0  0  0  12 237 1370 1062   79    0    0    0    0   0   0  0  0  0  0
 10: 0 0 0  0  0   5 111 1124 2055  588    2    0    0    0   0   0  0  0  0  0
 11: 0 0 0  0  0   2  39  684 2440 1686  127    0    0    0   0   0  0  0  0  0
 12: 0 0 0  0  0   0  15  348 2182 2782  641    2    0    0   0   0  0  0  0  0
 13: 0 0 0  0  0   0   4  163 1569 3252 1598   80    0    0   0   0  0  0  0  0
 14: 0 0 0  0  0   0   2   63  992 3217 2622  373    1    0   0   0  0  0  0  0
 15: 0 0 0  0  0   0   0   21  539 2731 3287  923   20    0   0   0  0  0  0  0
 16: 0 0 0  0  0   0   0    8  225 1999 3625 1623  105    0   0   0  0  0  0  0
 17: 0 0 0  0  0   0   0    2   96 1271 3380 2411  268    3   0   0  0  0  0  0
 18: 0 0 0  0  0   0   0    0   33  727 2864 2887  629   10   0   0  0  0  0  0
 19: 0 0 0  0  0   0   0    0    9  358 2123 3124 1054   36   0   0  0  0  0  0
 20: 0 0 0  0  0   0   0    0    2  142 1428 2912 1540  105   0   0  0  0  0  0
 21: 0 0 0  0  0   0   0    0    0   54  885 2527 1890  243   2   0  0  0  0  0
 22: 0 0 0  0  0   0   0    0    0   14  451 1936 2087  448   3   0  0  0  0  0
 23: 0 0 0  0  0   0   0    0    0    4  222 1394 2071  675  16   0  0  0  0  0
 24: 0 0 0  0  0   0   0    0    0    0   84  900 1791  925  47   0  0  0  0  0
 25: 0 0 0  0  0   0   0    0    0    0   20  536 1518 1064 106   1  0  0  0  0
 26: 0 0 0  0  0   0   0    0    0    0    5  265 1112 1124 192   0  0  0  0  0
 27: 0 0 0  0  0   0   0    0    0    0    0  132  784 1073 291   3  0  0  0  0
 28: 0 0 0  0  0   0   0    0    0    0    0   35  496  945 376  11  0  0  0  0
 29: 0 0 0  0  0   0   0    0    0    0    0    9  291  749 452  26  0  0  0  0
 30: 0 0 0  0  0   0   0    0    0    0    0    1  144  572 473  46  0  0  0  0
 31: 0 0 0  0  0   0   0    0    0    0    0    0   56  377 457  97  0  0  0  0
 32: 0 0 0  0  0   0   0    0    0    0    0    0   15  258 395 102  1  0  0  0
 33: 0 0 0  0  0   0   0    0    0    0    0    0    1  141 324 158  3  0  0  0
 34: 0 0 0  0  0   0   0    0    0    0    0    0    0   63 239 137  5  0  0  0
 35: 0 0 0  0  0   0   0    0    0    0    0    0    0   27 167 175 18  0  0  0
 36: 0 0 0  0  0   0   0    0    0    0    0    0    0    3 110 129 17  0  0  0
 37: 0 0 0  0  0   0   0    0    0    0    0    0    0    0  57 122 32  0  0  0
 38: 0 0 0  0  0   0   0    0    0    0    0    0    0    0  27  87 43  0  0  0
 39: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   6  62 31  2  0  0
 40: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0  47 47  4  0  0
 41: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0  19 26  0  0  0
 42: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0  10 29 10  0  0
 43: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   1 21  6  0  0
 44: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0 12  5  0  0
 45: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  7 13  0  0
 46: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  1  3  0  0
 47: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  8  2  0
 48: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  2  0  0
 49: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  3  0  0
 50: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  0  3  0
 51: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  0  0  0
 52: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  0  1  0
 53: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  0  0  0
 54: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  0  0  0
 55: 0 0 0  0  0   0   0    0    0    0    0    0    0    0   0   0  0  0  0  1

The total number is 113346.

The unique PARR configurations where all neighbour points have a connection, except for crossing diagonal connections where either one or the other is included, are:

     1 2 3  4  5   6   7    8    9   10   11    12    13    14    15    16   17   18   19   20
   +------------------------------------------------------------------------------------------
  0: 1 0 0  0  0   0   0    0    0    0    0     0     0     0     0     0    0    0    0    0
  1: 0 2 0  0  0   0   0    0    0    0    0     0     0     0     0     0    0    0    0    0
  2: 0 0 4  0  0   0   0    0    0    0    0     0     0     0     0     0    0    0    0    0
  3: 0 0 1 14  0   0   0    0    0    0    0     0     0     0     0     0    0    0    0    0
  4: 0 0 0  5 50   0   0    0    0    0    0     0     0     0     0     0    0    0    0    0
  5: 0 0 0  3 26 148   0    0    0    0    0     0     0     0     0     0    0    0    0    0
  6: 0 0 0  0 14 156 323    0    0    0    0     0     0     0     0     0    0    0    0    0
  7: 0 0 0  0  4  96 548  374    0    0    0     0     0     0     0     0    0    0    0    0
  8: 0 0 0  0  1  36 486 1048  269    0    0     0     0     0     0     0    0    0    0    0
  9: 0 0 0  0  0  17 299 1480 1062   79    0     0     0     0     0     0    0    0    0    0
 10: 0 0 0  0  0   2 136 1389 2203  588    2     0     0     0     0     0    0    0    0    0
 11: 0 0 0  0  0   0  44  967 3014 1766  127     0     0     0     0     0    0    0    0    0
 12: 0 0 0  0  0   0  10  463 3211 3392  665     2     0     0     0     0    0    0    0    0
 13: 0 0 0  0  0   0   0  180 2594 4957 1902    80     0     0     0     0    0    0    0    0
 14: 0 0 0  0  0   0   0   38 1538 5653 4011   443     1     0     0     0    0    0    0    0
 15: 0 0 0  0  0   0   0    4  654 5458 6284  1401    20     0     0     0    0    0    0    0
 16: 0 0 0  0  0   0   0    0  184 4114 7901  3364   154     0     0     0    0    0    0    0
 17: 0 0 0  0  0   0   0    0   20 2262 8607  6010   669     3     0     0    0    0    0    0
 18: 0 0 0  0  0   0   0    0    0  777 8033  8697  1837    20     0     0    0    0    0    0
 19: 0 0 0  0  0   0   0    0    0  150 5846 10512  4052   169     0     0    0    0    0    0
 20: 0 0 0  0  0   0   0    0    0    6 2870 11345  7088   597     0     0    0    0    0    0
 21: 0 0 0  0  0   0   0    0    0    0  779 10733  9833  1732    10     0    0    0    0    0
 22: 0 0 0  0  0   0   0    0    0    0   78  7416 12018  3844    94     0    0    0    0    0
 23: 0 0 0  0  0   0   0    0    0    0    0  3151 13518  6665   371     0    0    0    0    0
 24: 0 0 0  0  0   0   0    0    0    0    0   588 12907  9531  1165     0    0    0    0    0
 25: 0 0 0  0  0   0   0    0    0    0    0    27  8264 12081  2937    21    0    0    0    0
 26: 0 0 0  0  0   0   0    0    0    0    0     0  2760 14745  5139   131    0    0    0    0
 27: 0 0 0  0  0   0   0    0    0    0    0     0   312 14046  7711   668    0    0    0    0
 28: 0 0 0  0  0   0   0    0    0    0    0     0     0  7635 11038  1541    0    0    0    0
 29: 0 0 0  0  0   0   0    0    0    0    0     0     0  1870 14601  3294   10    0    0    0
 30: 0 0 0  0  0   0   0    0    0    0    0     0     0    64 13034  5432  208    0    0    0
 31: 0 0 0  0  0   0   0    0    0    0    0     0     0     0  5754  8800  611    0    0    0
 32: 0 0 0  0  0   0   0    0    0    0    0     0     0     0   676 12996 1673    0    0    0
 33: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0 10220 2882    0    0    0
 34: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0  2816 6580  116    0    0
 35: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0    72 9864  700    0    0
 36: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0     0 5904  996    0    0
 37: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0     0  512 4456    0    0
 38: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0     0    0 6144    0    0
 39: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0     0    0 1568  392    0
 40: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0     0    0    0 2576    0
 41: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0     0    0    0 2048    0
 42: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0     0    0    0    0    0
 43: 0 0 0  0  0   0   0    0    0    0    0     0     0     0     0     0    0    0    0 1056

The total number is 466446.


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