A320796 Regular triangle where T(n,k) is the number of non-isomorphic self-dual multiset partitions of weight n with k parts.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 5, 7, 3, 1, 1, 7, 14, 10, 3, 1, 1, 9, 23, 24, 11, 3, 1, 1, 12, 39, 53, 34, 12, 3, 1, 1, 14, 61, 102, 86, 39, 12, 3, 1, 1, 17, 90, 193, 201, 117, 42, 12, 3, 1, 1, 20, 129, 340, 434, 310, 136, 43, 12, 3, 1, 1, 24, 184, 584, 902, 778, 412, 149, 44, 12, 3, 1

Offset: 1

Author

Gus Wiseman, Nov 02 2018

Keywords

• nonn
• tabl

Comments

Also the number of nonnegative integer k X k symmetric matrices with sum of elements equal to n and no zero rows or columns, up to row and column permutations.

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Triangle begins:
   1
   1   1
   1   2   1
   1   4   3   1
   1   5   7   3   1
   1   7  14  10   3   1
   1   9  23  24  11   3   1
   1  12  39  53  34  12   3   1
   1  14  61 102  86  39  12   3   1
   1  17  90 193 201 117  42  12   3   1
Non-isomorphic representatives of the multiset partitions for n = 1 through 5 (commas elided):
1: {{1}}
.
2: {{11}}  {{1}{2}}
.
3: {{111}}  {{1}{22}}  {{1}{2}{3}}
.           {{2}{12}}
.
4: {{1111}}  {{11}{22}}  {{1}{1}{23}}  {{1}{2}{3}{4}}
.            {{12}{12}}  {{1}{2}{33}}
.            {{1}{222}}  {{1}{3}{23}}
.            {{2}{122}}
.
5: {{11111}}  {{11}{122}}  {{1}{22}{33}}  {{1}{2}{2}{34}}  {{1}{2}{3}{4}{5}}
.             {{11}{222}}  {{1}{23}{23}}  {{1}{2}{3}{44}}
.             {{12}{122}}  {{1}{2}{333}}  {{1}{2}{4}{34}}
.             {{1}{2222}}  {{1}{3}{233}}
.             {{2}{1222}}  {{2}{12}{33}}
.                          {{2}{13}{23}}
.                          {{3}{3}{123}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

Row sums are A316983.

Cf. A000219, A007716, A316980, A317533, A318805, A319560, A319616, A319721, A320797-A320813.

Programs

PARI

row(n)={vector(n, k, T(k,n) - T(k-1,n))} \\ T(n,k) defined in A318805. - Andrew Howroyd, Jan 16 2024

Formula

T(n,k) = A318805(k,n) - A318805(k-1,n). - Andrew Howroyd, Jan 16 2024

Extensions

a(56) onwards from Andrew Howroyd, Jan 16 2024