OFFSET
0,4
COMMENTS
The incidence matrix of a multiset partition has entry (i, j) equal to the multiplicity of vertex i in part j.
Also the number of positive integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with all different entries.
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..1000
FORMULA
a(n) = Sum_{k>=1} (A121860(k) + A121860(k+1) - 2)*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018
EXAMPLE
Non-isomorphic representatives of the a(3) = 3 through a(7) = 15 multiset partitions:
{{111}} {{1111}} {{11111}} {{111111}} {{1111111}}
{{122}} {{1222}} {{11222}} {{112222}} {{1112222}}
{{1}{11}} {{1}{111}} {{12222}} {{122222}} {{1122222}}
{{1}{1111}} {{122333}} {{1222222}}
{{11}{111}} {{1}{11111}} {{1223333}}
{{11}{1111}} {{1}{111111}}
{{1}{11222}} {{11}{11111}}
{{11}{1222}} {{111}{1111}}
{{112}{222}} {{1}{112222}}
{{122}{222}} {{11}{12222}}
{{2}{11222}} {{112}{2222}}
{{22}{1222}} {{122}{2222}}
{{1}{11}{111}} {{2}{112222}}
{{22}{12222}}
{{1}{11}{1111}}
MATHEMATICA
(* b = A121860 *) b[n_] := Sum[n!/(d! (n/d)!), {d, Divisors[n]}];
(* c = A008289 *) c[n_, k_] := c[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, c[n - k, k] + c[n - k, k - 1]]];
a[n_] := If[n == 0, 1, Sum[ (b[k] + b[k + 1] - 2) c[n, k], {k, 1, n}]];
a /@ Range[0, 45] (* Jean-François Alcover, Sep 14 2019 *)
PROG
(PARI) \\ here b(n) is A121860(n).
b(n)={sumdiv(n, d, n!/(d!*(n/d)!))}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, b(n-1)+b(n)-2))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 15 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018
STATUS
approved