A321403 - OEIS

%I #9 May 31 2023 22:35:40

%S 1,1,1,2,4,6,10,17,32,56,98,177,335,620,1164,2231,4349,8511,16870,

%T 33844,68746,140894,291698,610051,1288594,2745916,5903988,12805313,

%U 28010036,61764992,137281977,307488896,693912297,1577386813,3611241900,8324940862,19321470086

%N Number of non-isomorphic self-dual set multipartitions (multisets of sets) of weight n.

%C Also the number of symmetric (0,1)-matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns.

%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="/A321403/b321403.txt">Table of n, a(n) for n = 0..50</a>

%e Non-isomorphic representatives of the a(1) = 1 through a(7) = 17 set multipartitions:

%e   {{1}}  {{1},{2}}  {{2},{1,2}}    {{1,2},{1,2}}      {{1},{2,3},{2,3}}

%e                     {{1},{2},{3}}  {{1},{1},{2,3}}    {{2},{1,3},{2,3}}

%e                                    {{1},{3},{2,3}}    {{3},{3},{1,2,3}}

%e                                    {{1},{2},{3},{4}}  {{1},{2},{2},{3,4}}

%e                                                       {{1},{2},{4},{3,4}}

%e                                                       {{1},{2},{3},{4},{5}}

%e .

%e   {{1,2},{1,3},{2,3}}        {{1,3},{2,3},{1,2,3}}

%e   {{3},{2,3},{1,2,3}}        {{1},{1},{1,4},{2,3,4}}

%e   {{1},{1},{1},{2,3,4}}      {{1},{2,3},{2,4},{3,4}}

%e   {{1},{2},{3,4},{3,4}}      {{1},{4},{3,4},{2,3,4}}

%e   {{1},{3},{2,4},{3,4}}      {{2},{1,2},{3,4},{3,4}}

%e   {{1},{4},{4},{2,3,4}}      {{2},{1,3},{2,4},{3,4}}

%e   {{2},{4},{1,2},{3,4}}      {{3},{4},{1,4},{2,3,4}}

%e   {{1},{2},{3},{3},{4,5}}    {{4},{4},{4},{1,2,3,4}}

%e   {{1},{2},{3},{5},{4,5}}    {{1},{1},{5},{2,3},{4,5}}

%e   {{1},{2},{3},{4},{5},{6}}  {{1},{2},{2},{2},{3,4,5}}

%e                              {{1},{2},{3},{4,5},{4,5}}

%e                              {{1},{2},{4},{3,5},{4,5}}

%e                              {{1},{2},{5},{5},{3,4,5}}

%e                              {{1},{3},{5},{2,3},{4,5}}

%e                              {{1},{2},{3},{4},{4},{5,6}}

%e                              {{1},{2},{3},{4},{6},{5,6}}

%e                              {{1},{2},{3},{4},{5},{6},{7}}

%e Inequivalent representatives of the a(6) = 10 matrices:

%e   [0 0 1] [1 1 0]

%e   [0 1 1] [1 0 1]

%e   [1 1 1] [0 1 1]

%e .

%e   [1 0 0 0] [1 0 0 0] [1 0 0 0] [1 0 0 0] [0 1 0 0]

%e   [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] [0 0 0 1]

%e   [1 0 0 0] [0 0 1 1] [0 1 0 1] [0 0 0 1] [1 1 0 0]

%e   [0 1 1 1] [0 0 1 1] [0 0 1 1] [0 1 1 1] [0 0 1 1]

%e .

%e   [1 0 0 0 0] [1 0 0 0 0]

%e   [0 1 0 0 0] [0 1 0 0 0]

%e   [0 0 1 0 0] [0 0 1 0 0]

%e   [0 0 1 0 0] [0 0 0 0 1]

%e   [0 0 0 1 1] [0 0 0 1 1]

%e .

%e   [1 0 0 0 0 0]

%e   [0 1 0 0 0 0]

%e   [0 0 1 0 0 0]

%e   [0 0 0 1 0 0]

%e   [0 0 0 0 1 0]

%e   [0 0 0 0 0 1]

%o (PARI)

%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

%o c(p, k)={polcoef((prod(i=2, #p, prod(j=1, i-1, (1 + x^(2*lcm(p[i], p[j])) + O(x*x^k))^gcd(p[i], p[j]))) * prod(i=1, #p, my(t=p[i]); (1 + x^t + O(x*x^k))^(t%2)*(1 + x^(2*t) + O(x*x^k))^(t\2) )), k)}

%o a(n)={my(s=0); forpart(p=n, s+=permcount(p)*c(p, n)); s/n!} \\ _Andrew Howroyd_, May 31 2023

%Y Cf. A007716, A049311, A135588, A135589, A138178, A283877, A316983, A319616.

%Y Cf. A320796, A320797, A321403, A321404, A321405.

%K nonn

%O 0,4

%A _Gus Wiseman_, Nov 15 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, May 31 2023