A326026 Number of non-isomorphic multiset partitions of weight n where each part has a different length.

1, 1, 2, 7, 12, 35, 111, 247, 624, 1843, 6717, 15020, 46847, 124808, 412577, 1658973, 4217546, 12997734, 40786810, 126971940, 437063393, 2106317043, 5499108365, 19037901867, 59939925812, 210338815573, 683526043801, 2741350650705, 14848209030691, 41533835240731, 151548411269815

Offset: 0

Author

Gus Wiseman, Jul 13 2019

Keywords

• nonn

Comments

The number of non-isomorphic multiset partitions of weight n is A007716(n).

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions:
  {{1}}  {{1,1}}  {{1,1,1}}    {{1,1,1,1}}
         {{1,2}}  {{1,2,2}}    {{1,1,2,2}}
                  {{1,2,3}}    {{1,2,2,2}}
                  {{1},{1,1}}  {{1,2,3,3}}
                  {{1},{2,2}}  {{1,2,3,4}}
                  {{1},{2,3}}  {{1},{1,1,1}}
                  {{2},{1,2}}  {{1},{1,2,2}}
                               {{1},{2,2,2}}
                               {{1},{2,3,3}}
                               {{1},{2,3,4}}
                               {{2},{1,2,2}}
                               {{3},{1,2,3}}
		

Links

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

Crossrefs

Row sums of A332260.

Cf. A007716, A007837, A303546, A306017, A316980, A316983, A319560, A326514, A326517, A326533.

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
D(p,n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); polcoef(prod(k=1, #u, 1 + u[k]*x^k + O(x*x^n)), n)/prod(i=1, #v, i^v[i]*v[i]!)}
a(n)={my(s=0); forpart(p=n, s+=D(p,n)); s} \\ Andrew Howroyd, Feb 08 2020

Extensions

Terms a(11) and beyond from Andrew Howroyd, Feb 08 2020