A358721 -id:A358721 - cp's OEIS frontend

A322487 Number of (3*n) X n matrices with nonnegative integer entries and each column sum being 3 up to permutation of rows.

1, 3, 31, 686, 27036, 1688360, 154703688, 19692332568, 3342458334775, 732812082630803, 202322386045180686, 68898094282978653925, 28443422251718020038049, 14029033632468285836567998, 8164217197799501761637725983, 5545466507405459243366712102466

Offset: 0

Andrew Howroyd, Dec 11 2018

nonn

Also number of multiset partitions of [1,1,1,2,2,2,...,n,n,n] into nonempty multisets. - Marko Riedel, Nov 29 2022

			a(1) = 3 because up to permutations of rows there are 3 column vectors with sum 3: [1, 1, 1], [2, 1, 0] and [3, 0, 0].

Andrew Howroyd, Table of n, a(n) for n = 0..100

Row n=3 of A219727.

Cf. A165434, A358721.

A358722 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 1, 1, 2, 2, 2, 2, ..., n, n, n, n] into k nonempty submultisets, for 1 <= k <= 4n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 12, 29, 32, 21, 10, 3, 1, 1, 62, 513, 1399, 1857, 1513, 855, 364, 119, 31, 6, 1, 1, 312, 8165, 55704, 155989, 231642, 215250, 139789, 68154, 26135, 8105, 2071, 435, 75, 10, 1, 1, 1562, 125121, 2076531, 12235869, 34100001, 53914814, 54898626, 39436580, 21332108, 9098469, 3160761, 914625, 223740, 46628, 8291, 1245, 155, 15, 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=4.

			The triangular array starts:
[0]: 1
[1]: 1,  2,   1,    1;
[2]: 1, 12,  29,   32,   21,   10,   3,   1;
[3]: 1, 62, 513, 1399, 1857, 1513, 855, 364, 119, 31, 6, 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, A358721, A358781 (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.

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A322487 Number of (3*n) X n matrices with nonnegative integer entries and each column sum being 3 up to permutation of rows.

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A358722 Triangle read by rows. Number T(n, k) of partitions of the multiset [1, 1, 1, 1, 2, 2, 2, 2, ..., n, n, n, n] into k nonempty submultisets, for 1 <= k <= 4n.

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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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