A321403 Number of non-isomorphic self-dual set multipartitions (multisets of sets) of weight n.

1, 1, 1, 2, 4, 6, 10, 17, 32, 56, 98, 177, 335, 620, 1164, 2231, 4349, 8511, 16870, 33844, 68746, 140894, 291698, 610051, 1288594, 2745916, 5903988, 12805313, 28010036, 61764992, 137281977, 307488896, 693912297, 1577386813, 3611241900, 8324940862, 19321470086

Offset: 0

Author

Gus Wiseman, Nov 15 2018

Keywords

• nonn

Comments

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.

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

			Non-isomorphic representatives of the a(1) = 1 through a(7) = 17 set multipartitions:
  {{1}}  {{1},{2}}  {{2},{1,2}}    {{1,2},{1,2}}      {{1},{2,3},{2,3}}
                    {{1},{2},{3}}  {{1},{1},{2,3}}    {{2},{1,3},{2,3}}
                                   {{1},{3},{2,3}}    {{3},{3},{1,2,3}}
                                   {{1},{2},{3},{4}}  {{1},{2},{2},{3,4}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{3},{4},{5}}
.
  {{1,2},{1,3},{2,3}}        {{1,3},{2,3},{1,2,3}}
  {{3},{2,3},{1,2,3}}        {{1},{1},{1,4},{2,3,4}}
  {{1},{1},{1},{2,3,4}}      {{1},{2,3},{2,4},{3,4}}
  {{1},{2},{3,4},{3,4}}      {{1},{4},{3,4},{2,3,4}}
  {{1},{3},{2,4},{3,4}}      {{2},{1,2},{3,4},{3,4}}
  {{1},{4},{4},{2,3,4}}      {{2},{1,3},{2,4},{3,4}}
  {{2},{4},{1,2},{3,4}}      {{3},{4},{1,4},{2,3,4}}
  {{1},{2},{3},{3},{4,5}}    {{4},{4},{4},{1,2,3,4}}
  {{1},{2},{3},{5},{4,5}}    {{1},{1},{5},{2,3},{4,5}}
  {{1},{2},{3},{4},{5},{6}}  {{1},{2},{2},{2},{3,4,5}}
                             {{1},{2},{3},{4,5},{4,5}}
                             {{1},{2},{4},{3,5},{4,5}}
                             {{1},{2},{5},{5},{3,4,5}}
                             {{1},{3},{5},{2,3},{4,5}}
                             {{1},{2},{3},{4},{4},{5,6}}
                             {{1},{2},{3},{4},{6},{5,6}}
                             {{1},{2},{3},{4},{5},{6},{7}}
Inequivalent representatives of the a(6) = 10 matrices:
  [0 0 1] [1 1 0]
  [0 1 1] [1 0 1]
  [1 1 1] [0 1 1]
.
  [1 0 0 0] [1 0 0 0] [1 0 0 0] [1 0 0 0] [0 1 0 0]
  [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] [0 0 0 1]
  [1 0 0 0] [0 0 1 1] [0 1 0 1] [0 0 0 1] [1 1 0 0]
  [0 1 1 1] [0 0 1 1] [0 0 1 1] [0 1 1 1] [0 0 1 1]
.
  [1 0 0 0 0] [1 0 0 0 0]
  [0 1 0 0 0] [0 1 0 0 0]
  [0 0 1 0 0] [0 0 1 0 0]
  [0 0 1 0 0] [0 0 0 0 1]
  [0 0 0 1 1] [0 0 0 1 1]
.
  [1 0 0 0 0 0]
  [0 1 0 0 0 0]
  [0 0 1 0 0 0]
  [0 0 0 1 0 0]
  [0 0 0 0 1 0]
  [0 0 0 0 0 1]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

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

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

Programs

PARI

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

Extensions

Terms a(11) and beyond from Andrew Howroyd, May 31 2023