A384380 Irregular triangle read by rows: T(n,k) is the number of connected subsets of k faces (or polyforms) of the n-th Johnson solid, up to symmetries of that solid; 1 <= n <= 92, 1 <= k <= A242731(n).

2, 2, 3, 2, 1, 2, 2, 3, 3, 2, 1, 4, 4, 9, 14, 14, 9, 4, 1, 4, 4, 12, 20, 32, 30, 23, 11, 4, 1, 4, 4, 13, 29, 54, 75, 75, 55, 31, 12, 4, 1, 5, 5, 17, 43, 118, 285, 595, 992, 1320, 1348, 1045, 603, 262, 86, 22, 5, 1, 3, 4, 6, 7, 6, 3, 1, 3, 4, 7, 12, 17, 16, 9, 3, 1

Offset: 1

Author

Pontus von Brömssen and Peter Kagey, May 28 2025

Keywords

• nonn
• tabf
• fini

Comments

Two faces are connected if they share an edge.

Equivalently, T(n,k) is the number of connected induced k-vertex subgraphs of the 1-skeleton of the dual of the n-th Johnson solid, up to symmetries of that dual.

Examples

			Triangle begins:
  n\k| 1  2  3  4   5   6   7    8    9   10   11   12   13  14  15 16 17 18
  ---+----------------------------------------------------------------------
   1 | 2  2  3  2   1
   2 | 2  2  3  3   2   1
   3 | 4  4  9 14  14   9   4    1
   4 | 4  4 12 20  32  30  23   11    4    1
   5 | 4  4 13 29  54  75  75   55   31   12    4    1
   6 | 5  5 17 43 118 285 595  992 1320 1348 1045  603  262  86  22  5  1
   7 | 3  4  6  7   6   3   1
   8 | 3  4  7 12  17  16   9    3    1
   9 | 3  4  7 13  24  35  36   22    9    3    1
  10 | 4  4  8 15  28  47  81  102   87   45   16    4    1
  11 | 4  4  8 17  35  71 139  252  378  429  340  183   67  18   4  1
  12 | 1  2  2  3   1   1
  13 | 1  2  2  5   6  10   7    5    1    1
  14 | 2  3  5  7  10   9   6    2    1
  15 | 2  3  5 11  19  31  38   38   20   10    2    1
  16 | 2  3  5 11  23  45  82  126  154  130   77   30   10   2   1
  17 | 2  3  4 10  16  35  61  120  180  237  194  117   40  13   2  1
  18 | 6  7 17 36  81 165 300  386  337  197   82   25    6   1
  19 | 6  7 20 44 121 290 701 1403 2359 3047 2975 2110 1106 435 131 31  6  1
		

Links

• Pontus von Brömssen, Table of n, a(n) for n = 1..341 (first 24 rows)
• Eric Weisstein's World of Mathematics, Johnson Solid.
• Wikipedia, List of Johnson solids.

Crossrefs

Cf. A242731 (row lengths), A384376, A384378, A384381 (row sums).