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A320810 Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1. 9
0, 2, 3, 12, 7, 84, 15, 410, 354, 3073 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..10.

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}}

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,more

AUTHOR

Gus Wiseman, Nov 15 2018

STATUS

approved

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Last modified July 10 00:25 EDT 2020. Contains 335570 sequences. (Running on oeis4.)