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A326517 Number of normal multiset partitions of weight n where each part has a different size. 12
1, 1, 2, 12, 28, 140, 956, 3520, 17792, 111600, 1144400, 4884064, 34907936, 214869920, 1881044032, 25687617152, 139175009920, 1098825972608, 8770328141888, 74286112885504, 784394159958848, 15114871659653952, 92392468773724544, 889380453354852416, 7652770202041529856 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

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

EXAMPLE

The a(0) = 1 through a(3) = 12 normal multiset partitions:

  {}  {{1}}  {{1,1}}  {{1,1,1}}

             {{1,2}}  {{1,1,2}}

                      {{1,2,2}}

                      {{1,2,3}}

                      {{1},{1,1}}

                      {{1},{1,2}}

                      {{1},{2,2}}

                      {{1},{2,3}}

                      {{2},{1,1}}

                      {{2},{1,2}}

                      {{2},{1,3}}

                      {{3},{1,2}}

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_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];

Table[Length[Select[Join@@mps/@allnorm[n], UnsameQ@@Length/@#&]], {n, 0, 6}]

PROG

(PARI)

R(n, k)={Vec(prod(j=1, n, 1 + binomial(k+j-1, j)*x^j + O(x*x^n)))}

seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ Andrew Howroyd, Feb 07 2020

CROSSREFS

Row sums of A332253.

Cf. A007837, A038041, A255906, A317583, A326026, A326514, A326518, A326519, A326520, A326521, A326533.

Sequence in context: A034318 A338798 A345694 * A248119 A240764 A215784

Adjacent sequences:  A326514 A326515 A326516 * A326518 A326519 A326520

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 12 2019

EXTENSIONS

Terms a(8) and beyond from Andrew Howroyd, Feb 07 2020

STATUS

approved

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Last modified January 17 11:57 EST 2022. Contains 350394 sequences. (Running on oeis4.)