OFFSET
0,3
COMMENTS
A multiset is uniform if all multiplicities are equal.
Also the number of factorizations into factors > 1 of primorial powers k in A100778 with sum of prime indices A056239(k) equal to n.
a(n) is the number of nonequivalent nonnegative integer matrices without zero rows or columns with equal column sums and total sum n up to permutation of rows. - Andrew Howroyd, Jan 11 2020
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..50
FORMULA
a(n) = Sum_{d|n} A219727(n/d, d). - Andrew Howroyd, Jan 11 2020
EXAMPLE
The a(1) = 1 through a(4) = 29 multiset partitions:
{{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
{{1,2}} {{1,2,3}} {{1,1,2,2}}
{{1},{1}} {{1},{1,1}} {{1,2,3,4}}
{{1},{2}} {{1},{2,3}} {{1},{1,1,1}}
{{2},{1,3}} {{1,1},{1,1}}
{{3},{1,2}} {{1},{1,2,2}}
{{1},{1},{1}} {{1,1},{2,2}}
{{1},{2},{3}} {{1,2},{1,2}}
{{1},{2,3,4}}
{{1,2},{3,4}}
{{1,3},{2,4}}
{{1,4},{2,3}}
{{2},{1,1,2}}
{{2},{1,3,4}}
{{3},{1,2,4}}
{{4},{1,2,3}}
{{1},{1},{1,1}}
{{1},{1},{2,2}}
{{1},{2},{1,2}}
{{1},{2},{3,4}}
{{1},{3},{2,4}}
{{1},{4},{2,3}}
{{2},{2},{1,1}}
{{2},{3},{1,4}}
{{2},{4},{1,3}}
{{3},{4},{1,2}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
{{1},{2},{3},{4}}
MATHEMATICA
u[n_, k_]:=u[n, k]=If[n==1, 1, Sum[u[n/d, d], {d, Select[Rest[Divisors[n]], #<=k&]}]];
Table[Sum[u[Array[Prime, d, 1, Times]^(n/d), Array[Prime, d, 1, Times]^(n/d)], {d, Divisors[n]}], {n, 12}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 26 2018
EXTENSIONS
a(14)-a(15) from Alois P. Heinz, Jan 16 2019
Terms a(16) and beyond from Andrew Howroyd, Jan 11 2020
STATUS
approved