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A317583
Number of multiset partitions of normal multisets of size n such that all blocks have the same size.
16
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
OFFSET
1,2
COMMENTS
A multiset is normal if it spans an initial interval of positive integers.
a(n) is the number of nonnegative integer matrices with total sum n, nonzero rows and each column with the same sum with columns in nonincreasing lexicographic order. - Andrew Howroyd, Jan 15 2020
FORMULA
a(p) = 2^p for prime p. - Andrew Howroyd, Sep 15 2018
a(n) = Sum_{d|n} A331315(n/d, d). - Andrew Howroyd, Jan 15 2020
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 01 2018
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Sep 15 2018
STATUS
approved