A306018 Number of non-isomorphic set multipartitions of weight n in which all parts have the same size.

1, 1, 3, 4, 9, 8, 24, 16, 51, 47, 115, 57, 420, 102, 830, 879, 2962, 298, 15527, 491, 41275, 80481, 133292, 1256, 2038182, 58671, 2386862, 24061887, 23570088, 4566, 600731285, 6843, 1303320380, 14138926716, 1182784693, 1820343112, 542834549721, 21638, 31525806080

Offset: 0

Author

Gus Wiseman, Jun 17 2018

Keywords

• nonn

Comments

A set multipartition of weight n is a finite multiset of finite nonempty sets whose cardinalities sum to n.

Number of distinct binary matrices with all row sums equal and total sum n, up to row and column permutations. - Andrew Howroyd, Sep 05 2018

Examples

			Non-isomorphic representatives of the a(6) = 24 set multipartitions in which all parts have the same size:
{{1,2,3,4,5,6}}
{{1,2,3},{1,2,3}}
{{1,2,3},{4,5,6}}
{{1,2,5},{3,4,5}}
{{1,3,4},{2,3,4}}
{{1,2},{1,2},{1,2}}
{{1,2},{1,3},{2,3}}
{{1,2},{3,4},{3,4}}
{{1,2},{3,4},{5,6}}
{{1,2},{3,5},{4,5}}
{{1,3},{2,3},{2,3}}
{{1,3},{2,4},{3,4}}
{{1,4},{2,4},{3,4}}
{{1},{1},{1},{1},{1},{1}}
{{1},{1},{1},{2},{2},{2}}
{{1},{1},{2},{2},{2},{2}}
{{1},{1},{2},{2},{3},{3}}
{{1},{2},{2},{2},{2},{2}}
{{1},{2},{2},{3},{3},{3}}
{{1},{2},{3},{3},{3},{3}}
{{1},{2},{3},{3},{4},{4}}
{{1},{2},{3},{4},{4},{4}}
{{1},{2},{3},{4},{5},{5}}
{{1},{2},{3},{4},{5},{6}}
		

Links

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

Crossrefs

Cf. A000005, A000041, A001315, A007716, A038041, A049311, A283877, A298422, A304942, A306017, A306019, A306020, A306021, A331461.

Programs

PARI

\\ See A304942 for Blocks
a(n)={sumdiv(n, d, Blocks(n/d, n, d))} \\ Andrew Howroyd, Sep 05 2018

Formula

a(p) = A000041(p) + 1 for prime p. - Andrew Howroyd, Sep 06 2018

a(n) = Sum_{d|n} A331461(n/d, d). - Andrew Howroyd, Feb 09 2020

Extensions

Terms a(11) and beyond from Andrew Howroyd, Sep 05 2018