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A358835 Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums. 3
1, 1, 3, 4, 8, 8, 17, 16, 31, 34, 54, 57, 108, 102, 166, 191, 294, 298, 504, 491, 803, 843, 1251, 1256, 2167, 1974, 3133, 3226, 4972, 4566, 8018, 6843, 11657, 11044, 17217, 15010, 28422, 21638, 38397, 35067, 58508, 44584, 91870, 63262, 125114, 106264, 177483 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{d|n} Sum_{j=1..n/d} binomial(d + A008284(n/d, j) - 1, d) for n > 0. - Andrew Howroyd, Dec 31 2022
EXAMPLE
The a(1) = 1 through a(6) = 17 multiset partitions:
{1} {2} {3} {4} {5} {6}
{11} {12} {13} {14} {15}
{1}{1} {111} {22} {23} {24}
{1}{1}{1} {112} {113} {33}
{1111} {122} {114}
{2}{2} {1112} {123}
{11}{11} {11111} {222}
{1}{1}{1}{1} {1}{1}{1}{1}{1} {1113}
{1122}
{3}{3}
{11112}
{111111}
{12}{12}
{2}{2}{2}
{111}{111}
{11}{11}{11}
{1}{1}{1}{1}{1}{1}
MATHEMATICA
Table[If[n==0, 1, Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d], n/d], SameQ@@Length/@#&], {d, Divisors[n]}]]]], {n, 0, 20}]
PROG
(PARI)
P(n, y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))
seq(n) = {my(u=Vec(P(n, y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, binomial(d + polcoef(p, j, y) - 1, d)))))} \\ Andrew Howroyd, Dec 31 2022
CROSSREFS
For just constant sums we have A305551, ranked by A326534.
For just constant lengths we have A319066, ranked by A320324.
The version for set partitions is A327899.
For distinct instead of constant lengths and sums we have A358832.
The version for twice-partitions is A358833.
A001970 counts multiset partitions of integer partitions.
A063834 counts twice-partitions, strict A296122.
Sequence in context: A310016 A030014 A047968 * A358833 A322117 A181778
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 05 2022
EXTENSIONS
Terms a(41) and beyond from Andrew Howroyd, Dec 31 2022
STATUS
approved

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Last modified April 24 05:49 EDT 2024. Contains 371918 sequences. (Running on oeis4.)