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A317583 Number of multiset partitions of normal multisets of size n such that all blocks have the same size. 12
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

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

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. A038041, A255906, A298422, A306017, A306019, A306020, A306021, A320324, A322794, A326517, A326518, A326519, A326520, A326521.

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 August 20 21:29 EDT 2019. Contains 326155 sequences. (Running on oeis4.)