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

0,3

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

Links

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

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

Example

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}}

Prog

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

Crossrefs

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

Sequence in context: A317715 A372121 A305551 * A076120 A082188 A202499

Adjacent sequences: A306015 A306016 A306017 * A306019 A306020 A306021

Keyword

nonn

Author

Gus Wiseman, Jun 17 2018

Extensions

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

Status

approved