A360039 -id:A360039 - cp's OEIS frontend

A360037 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 subsets, for 3 <= k <= 3n.

1, 1, 1, 1, 1, 1, 4, 10, 13, 7, 3, 1, 1, 14, 92, 221, 249, 172, 81, 25, 6, 1, 1, 50, 872, 4277, 8806, 9840, 6945, 3377, 1206, 325, 65, 10, 1, 1, 186, 8496, 85941, 320320, 585960, 627838, 442321, 221475, 82985, 24038, 5496, 995, 140, 15, 1

Offset: 1

Marko Riedel, Jan 22 2023

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:
[1]: 1;
[2]: 1,  1,  1,   1;
[3]: 1,  4, 10,  13,   7,   3,  1;
[4]: 1, 14, 92, 221, 249, 172, 81, 25, 6, 1;

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

Marko Riedel, Rows 1 to 10 of triangle, flattened.
Marko Riedel, Documentation of the algorithm used in the Maple code.
Marko Riedel, Maple code for sequence by plain enumeration and the Polya Enumeration Theorem, by substitution and by recurrence.

Row sums are A165434.

Cf. A098233, A360038, A360039.

read "a360037maple":  # see link
A360037Row := n -> seq(T2(n, k, 3), k = 3..n*3): seq(A360037Row(n), n = 1..6);

A360038 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 subsets, for 4 <= k <= 4n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 11, 19, 22, 13, 7, 3, 1, 1, 14, 117, 445, 873, 1002, 805, 483, 226, 81, 25, 6, 1, 1, 51, 1387, 12567, 47986, 96620, 120970, 104942, 67901, 34385, 14150, 4817, 1371, 325, 65, 10, 1, 1, 201, 18171, 396571, 3053216, 11003801, 22360580, 29114463, 26607981, 18227245, 9816458, 4301588, 1572206, 487670, 129880, 29828, 5901, 995, 140, 15, 1

Offset: 1

Marko Riedel, Jan 22 2023

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:
[1]: 1;
[2]: 1,  1,   1,   1,   1;
[3]: 1,  4,  11,  19,  22,   13,   7,   3,   1;
[4]: 1, 14, 117, 445, 873, 1002, 805, 483, 226, 81, 25, 6, 1;

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

Marko Riedel, Rows 1 to 9 of triangle, flattened.

Cf. A098233, A360037, A360039, A165435 (row sums).

A098233 Consider the family of ordinary multigraphs. Sequence gives the triangle read by rows giving coefficients of polynomials arising from enumeration of those multigraphs on n edges.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 7, 3, 1, 1, 13, 46, 47, 25, 6, 1, 1, 40, 295, 587, 516, 235, 65, 10, 1, 1, 121, 1846, 6715, 9690, 7053, 3006, 800, 140, 15, 1, 1, 364, 11347, 73003, 170051, 189458, 119211, 46795, 12201, 2170, 266, 21, 1, 1, 1093, 68986, 768747

Offset: 0

N. J. A. Sloane, Oct 26 2004

Also gives number T(n, k) of partitions of the multiset {1, 1, 2, 2, ..., n, n} into k nonempty subsets, for 2 <= k <= 2n. - Marko Riedel, Jan 22 2023

			The first few polynomials are:
  1,
  x^2,
  x^2+x^3+x^4,
  x^2+4x^3+7x^4+3x^5+x^6,
  x^2+13x^3+46x^4+47x^5+25x^6+6x^7+x^8,
  x^2+40x^3+295x^4+587x^5+516x^6+235x^7+65x^8+10x^9+x^10,
  ...
Triangle starts:
  1;
  1;
  1,  1,   1;
  1,  4,   7,   3,   1;
  1, 13,  46,  47,  25,   6,  1;
  1, 40, 295, 587, 516, 235, 65, 10, 1;
  ...

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Steve Butler, Fan Chung, Jay Cummings, and R. L. Graham, Juggling card sequences, arXiv:1504.01426 [math.CO], 2015.
L. Comtet, Birecouvrements et birevêtements d'un ensemble fini, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Cf. A360037, A360038, A360039, A020554 (row sums).

cp's OEIS Frontend

A360037 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 subsets, for 3 <= k <= 3n.

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A360038 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 subsets, for 4 <= k <= 4n.

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A098233 Consider the family of ordinary multigraphs. Sequence gives the triangle read by rows giving coefficients of polynomials arising from enumeration of those multigraphs on n edges.

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Author

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Comments

Examples

References

Links

Crossrefs