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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.
19
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
1,5
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
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
FORMULA
T(n,k) = A318805(k,n) - A318805(k-1,n). - Andrew Howroyd, Jan 16 2024
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}}
PROG
(PARI) row(n)={vector(n, k, T(k, n) - T(k-1, n))} \\ T(n, k) defined in A318805. - Andrew Howroyd, Jan 16 2024
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Nov 02 2018
EXTENSIONS
a(56) onwards from Andrew Howroyd, Jan 16 2024
STATUS
approved