A384376 -id:A384376 - cp's OEIS frontend

A384378 Irregular triangle read by rows: T(n,k) is the number of connected subsets of k edges (or polysticks) of the n-th Johnson solid, up to symmetries of that solid; 1 <= n <= 92, 1 <= k <= A242732(n).

2, 4, 7, 12, 10, 6, 2, 1, 2, 4, 9, 17, 28, 25, 16, 7, 2, 1, 4, 7, 20, 47, 123, 274, 531, 779, 758, 504, 241, 87, 22, 4, 1, 4, 7, 20, 51, 144, 382, 990, 2332, 4873, 8546, 11776, 11733, 8529, 4673, 1957, 639, 156, 31, 4, 1

Offset: 1

Pontus von Brömssen and Peter Kagey, May 28 2025

			Triangle begins:
  n\k| 1  2  3  4   5   6   7    8    9   10    11    12   13   14   15  16  17 18 19 20
  ---+----------------------------------------------------------------------------------
  1  | 2  4  7 12  10   6   2    1
  2  | 2  4  9 17  28  25  16    7    2    1
  3  | 4  7 20 47 123 274 531  779  758  504   241    87   22    4    1
  4  | 4  7 20 51 144 382 990 2332 4873 8546 11776 11733 8529 4673 1957 639 156 31  4  1

Pontus von Brömssen, Table of n, a(n) for n = 1..78 (first 5 rows)
Eric Weisstein's World of Mathematics, Johnson Solid.
Wikipedia, List of Johnson solids.

Cf. A242732 (row lengths), A384376, A384379 (row sums), A384380.

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).

Original entry on oeis.org

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

Pontus von Brömssen and Peter Kagey, May 28 2025

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.

			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

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.

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

A384377 Number of connected induced subgraphs of the 1-skeleton of the n-th Johnson solid, up to symmetries of that solid.

Original entry on oeis.org

10, 13, 62, 224, 854, 14090, 30, 72, 157, 81, 196, 10, 20, 31, 79, 183, 86, 2196, 32069, 489168, 9965938, 3049, 49396, 843450, 17456717, 36, 261, 2117, 2019, 18717, 18302, 727214, 740170, 7545797, 9124, 9139, 314965, 11097938, 11101335

Offset: 1

Pontus von Brömssen and Peter Kagey, May 28 2025

The null-graph is not included in the counts.

Eric Weisstein's World of Mathematics, Johnson Solid.
Wikipedia, List of Johnson solids.

Row sums of A384376.

Cf. A384379, A384381.

cp's OEIS Frontend

A384378 Irregular triangle read by rows: T(n,k) is the number of connected subsets of k edges (or polysticks) of the n-th Johnson solid, up to symmetries of that solid; 1 <= n <= 92, 1 <= k <= A242732(n).

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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).

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A384377 Number of connected induced subgraphs of the 1-skeleton of the n-th Johnson solid, up to symmetries of that solid.

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Author

Keywords

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Links

Crossrefs