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

0,4

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.

Links

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

Example

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]

Prog

(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

Crossrefs

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

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

Sequence in context: A303430 A144023 A018164 * A340311 A025052 A142584

Adjacent sequences: A321400 A321401 A321402 * A321404 A321405 A321406

Keyword

nonn

Author

Gus Wiseman, Nov 15 2018

Extensions

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

Status

approved