A320796 - OEIS

%I #15 Jan 16 2024 19:52:41

%S 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,

%T 53,34,12,3,1,1,14,61,102,86,39,12,3,1,1,17,90,193,201,117,42,12,3,1,

%U 1,20,129,340,434,310,136,43,12,3,1,1,24,184,584,902,778,412,149,44,12,3,1

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

%C 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.

%C 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}}.

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

%H Andrew Howroyd, <a href="/A320796/b320796.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%F T(n,k) = A318805(k,n) - A318805(k-1,n). - _Andrew Howroyd_, Jan 16 2024

%e Triangle begins:

%e    1

%e    1   1

%e    1   2   1

%e    1   4   3   1

%e    1   5   7   3   1

%e    1   7  14  10   3   1

%e    1   9  23  24  11   3   1

%e    1  12  39  53  34  12   3   1

%e    1  14  61 102  86  39  12   3   1

%e    1  17  90 193 201 117  42  12   3   1

%e Non-isomorphic representatives of the multiset partitions for n = 1 through 5 (commas elided):

%e 1: {{1}}

%e .

%e 2: {{11}}  {{1}{2}}

%e .

%e 3: {{111}}  {{1}{22}}  {{1}{2}{3}}

%e .           {{2}{12}}

%e .

%e 4: {{1111}}  {{11}{22}}  {{1}{1}{23}}  {{1}{2}{3}{4}}

%e .            {{12}{12}}  {{1}{2}{33}}

%e .            {{1}{222}}  {{1}{3}{23}}

%e .            {{2}{122}}

%e .

%e 5: {{11111}}  {{11}{122}}  {{1}{22}{33}}  {{1}{2}{2}{34}}  {{1}{2}{3}{4}{5}}

%e .             {{11}{222}}  {{1}{23}{23}}  {{1}{2}{3}{44}}

%e .             {{12}{122}}  {{1}{2}{333}}  {{1}{2}{4}{34}}

%e .             {{1}{2222}}  {{1}{3}{233}}

%e .             {{2}{1222}}  {{2}{12}{33}}

%e .                          {{2}{13}{23}}

%e .                          {{3}{3}{123}}

%o (PARI) row(n)={vector(n, k, T(k,n) - T(k-1,n))} \\ T(n,k) defined in A318805. - _Andrew Howroyd_, Jan 16 2024

%Y Row sums are A316983.

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

%K nonn,tabl

%O 1,5

%A _Gus Wiseman_, Nov 02 2018

%E a(56) onwards from _Andrew Howroyd_, Jan 16 2024