A358722 -id:A358722 - cp's OEIS frontend

A358721 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 1, 2, 2, 2, ..., n, n, n] into k nonempty submultisets, for 1 <= k <= 3n.

1, 1, 1, 1, 1, 7, 11, 8, 3, 1, 1, 31, 139, 219, 175, 86, 28, 6, 1, 1, 127, 1547, 5321, 8004, 6687, 3579, 1329, 359, 71, 10, 1, 1, 511, 16171, 118605, 333887, 472784, 398771, 223700, 89640, 26853, 6171, 1100, 150, 15, 1, 1, 2047, 164651, 2511653, 13045458, 31207637, 41429946, 34621129, 19882236, 8342411, 2668319, 669446, 134075, 21591, 2785, 281, 21, 1

Offset: 0

Marko Riedel, Nov 28 2022

A generalization of ordinary Stirling set numbers to multisets that contain some m instances each of n elements, here we have m=3.

			The triangular array starts:
[0]: 1,
[1]: 1,   1,    1;
[2]: 1,   7,   11,    8,    3,    1;
[3]: 1,  31,  139,  219,  175,   86,   28,    6,   1;
[4]: 1, 127, 1547, 5321, 8004, 6687, 3579, 1329, 359, 71, 10, 1;

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Number of ways to partition a multiset into k non-empty multisets, Mathematics Stack Exchange.
Marko Riedel, Maple code for sequence by plain enumeration, the Polya Enumeration Theorem, and Power Group Enumeration

Cf. A008277, A358710, A358722, A322487 (row sums).

A358710 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 2, 2, ..., n, n] into k nonempty submultisets, for 1 <= k <= 2n.

Original entry on oeis.org

1, 1, 1, 1, 4, 3, 1, 1, 13, 26, 19, 6, 1, 1, 40, 183, 259, 163, 55, 10, 1, 1, 121, 1190, 3115, 3373, 1896, 620, 125, 15, 1, 1, 364, 7443, 34891, 62240, 54774, 27610, 8706, 1795, 245, 21, 1, 1, 1093, 45626, 374059, 1072316, 1435175, 1063570, 485850, 146363, 30261, 4361, 434, 28, 1

Offset: 0

Marko Riedel, Nov 27 2022

A generalization of ordinary Stirling set numbers to multisets that contain some m instances each of n elements, here we have m=2.

			The triangular array starts:
[0] 1;
[1] 1,   1;
[2] 1,   4,    3,     1;
[3] 1,  13,   26,    19,     6,     1;
[4] 1,  40,  183,   259,   163,    55,    10,    1;
[5] 1, 121, 1190,  3115,  3373,  1896,   620,  125,   15,   1;
[6] 1, 364, 7443, 34891, 62240, 54774, 27610, 8706, 1795, 245, 21, 1;

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Sidney Cadot, Table of n, a(n) for n = 0..420 (terms 1..420 from Marko Riedel)
Marko Riedel et al., Number of ways to partition a multiset into k non-empty multisets, Mathematics Stack Exchange.
Marko Riedel, Maple code for sequence by plain enumeration, the Polya Enumeration Theorem, and Power Group Enumeration.

Cf. A008277, A020555 (row sums), A358721, A358722.

A358781 Number of multiset partitions of [1,1,1,1,2,2,2,2,...,n,n,n,n] into nonempty multisets.

Original entry on oeis.org

1, 5, 109, 6721, 911838, 231575143, 99003074679, 66106443797808, 65197274052335504, 90954424202936106523, 173398227073956386079670, 439196881673194611574668282, 1443741072199958276777413001395

Offset: 0

Marko Riedel, Nov 29 2022

nonn

Generalization of Bell numbers to multiset partitions with m instances each of n different elements, here m=4.

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Number of ways to partition a multiset into k non-empty multisets, Mathematics Stack Exchange.

Cf. A358722, A020555, A322487.

Row n=4 of A219727.

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A358721 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 1, 2, 2, 2, ..., n, n, n] into k nonempty submultisets, for 1 <= k <= 3n.

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A358710 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 2, 2, ..., n, n] into k nonempty submultisets, for 1 <= k <= 2n.

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A358781 Number of multiset partitions of [1,1,1,1,2,2,2,2,...,n,n,n,n] into nonempty multisets.

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