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A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size. 4
1, 4, 8, 30, 32, 342, 128, 3754, 11360, 56138, 2048, 3834670, 8192, 27528494, 577439424, 2681075210, 131072, 238060300946, 524288, 11045144602614, 115488471132032, 49840258213638, 8388608, 152185891301461434, 140102945910265344, 124260001149229146, 85092642310351607968 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A multiset is normal if it spans an initial interval of positive integers.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200

FORMULA

a(p) = 2^p for prime p. - Andrew Howroyd, Sep 15 2018

EXAMPLE

The a(3) = 8 multiset partitions:

  {{1,1,1}}

  {{1,1,2}}

  {{1,2,2}}

  {{1,2,3}}

  {{1},{1},{1}}

  {{1},{1},{2}}

  {{1},{2},{2}}

  {{1},{2},{3}}

MATHEMATICA

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

allnorm[n_]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];

Table[Length[Select[Join@@mps/@allnorm[n], SameQ@@Length/@#&]], {n, 8}]

PROG

(PARI) \\ here U(n, m) gives number for m blocks of size n.

U(n, m)={sum(k=1, n*m, binomial(binomial(k+n-1, n)+m-1, m)*sum(r=k, n*m, binomial(r, k)*(-1)^(r-k)) )}

a(n)={sumdiv(n, d, U(d, n/d))} \\ Andrew Howroyd, Sep 15 2018

CROSSREFS

Cf. A000005, A007716, A038041, A255906, A298422, A306017, A306018, A306019, A306020, A306021, A317584.

Sequence in context: A297638 A256456 A270522 * A020331 A248476 A082595

Adjacent sequences:  A317580 A317581 A317582 * A317584 A317585 A317586

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 01 2018

EXTENSIONS

Terms a(9) and beyond from Andrew Howroyd, Sep 15 2018

STATUS

approved

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Last modified May 21 17:10 EDT 2019. Contains 323444 sequences. (Running on oeis4.)