A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.

0, 2, 3, 12, 7, 84, 15, 410, 354, 3073, 56, 28300, 101, 210036, 126839, 2070047, 297, 25295952, 490, 269662769, 89071291, 3449056162, 1255, 51132696310, 400625539, 713071048480, 145126661415, 11351097702297, 4565, 199926713003444, 6842, 3460838122540969

Offset

1,2

Comments

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the column sums are not relatively prime.

Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is periodic, where a multiset is periodic if its multiplicities have a common divisor > 1.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..50

Formula

a(n) = A007716(n) - A321283(n). - Andrew Howroyd, Jan 17 2023

Example

Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions whose part-sizes have a common divisor:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}
                      {{1,1},{2,2}}  {{1,2,3,4,5}}
                      {{1,2},{1,2}}
                      {{1,2},{2,2}}
                      {{1,2},{3,3}}
                      {{1,2},{3,4}}
                      {{1,3},{2,3}}
Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions with periodic multiset union:
  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
  {{1},{1}}  {{1},{1,1}}    {{1,1,2,2}}        {{1},{1,1,1,1}}
             {{1},{1},{1}}  {{1},{1,1,1}}      {{1,1},{1,1,1}}
                            {{1,1},{1,1}}      {{1},{1},{1,1,1}}
                            {{1},{1,2,2}}      {{1},{1,1},{1,1}}
                            {{1,1},{2,2}}      {{1},{1},{1},{1,1}}
                            {{1,2},{1,2}}      {{1},{1},{1},{1},{1}}
                            {{1},{1},{1,1}}
                            {{1},{1},{2,2}}
                            {{1},{2},{1,2}}
                            {{1},{1},{1},{1}}
                            {{1},{1},{2},{2}}

Prog

(PARI) \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); Vec(OgfSeries(sCartProd(sExp(A), -sum(d=2, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A, i*d)*x^(i*d)))) ))), -n)} \\ Andrew Howroyd, Jan 17 2023

Crossrefs

Cf. A007716, A018783, A047966, A120733, A303546, A303547, A305563, A319149, A319162, A319164, A319810.

Sequence in context: A282302 A282075 A282514 * A104038 A112979 A225723

Adjacent sequences: A320807 A320808 A320809 * A320811 A320812 A320813

Keyword

nonn

Author

Gus Wiseman, Nov 15 2018

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

Status

approved