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A320796 Regular triangle where T(n,k) is the number of non-isomorphic self-dual multiset partitions of weight n with k parts. 17
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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.

LINKS

Table of n, a(n) for n=1..55.

EXAMPLE

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}}

CROSSREFS

Row sums are A316983.

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

Sequence in context: A112096 A217874 A229118 * A026725 A026758 A130523

Adjacent sequences:  A320793 A320794 A320795 * A320797 A320798 A320799

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Nov 02 2018

STATUS

approved

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Last modified December 17 09:43 EST 2018. Contains 318193 sequences. (Running on oeis4.)