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A320807 Number of non-isomorphic multiset partitions of weight n in which all parts are aperiodic and all parts of the dual are also aperiodic. 1
1, 1, 3, 6, 17, 41, 122, 345, 1077, 3385, 11214 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also the number of nonnegative integer matrices up to row and column permutations with sum of entries equal to n and no zero rows or columns, in which each row and each column has relatively prime nonzero entries.

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

A multiset is aperiodic if its multiplicities are relatively prime.

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=0..10.

EXAMPLE

Non-isomorphic representatives of the a(1) = 1 through a(4) = 17 multiset partitions:

  {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}

         {{1},{1}}  {{1},{2,3}}    {{1,2},{1,2}}

         {{1},{2}}  {{2},{1,2}}    {{1},{2,3,4}}

                    {{1},{1},{1}}  {{1,2},{3,4}}

                    {{1},{2},{2}}  {{1,3},{2,3}}

                    {{1},{2},{3}}  {{2},{1,2,2}}

                                   {{3},{1,2,3}}

                                   {{1},{1},{2,3}}

                                   {{1},{2},{1,2}}

                                   {{1},{2},{3,4}}

                                   {{1},{3},{2,3}}

                                   {{2},{2},{1,2}}

                                   {{1},{1},{1},{1}}

                                   {{1},{1},{2},{2}}

                                   {{1},{2},{2},{2}}

                                   {{1},{2},{3},{3}}

                                   {{1},{2},{3},{4}}

CROSSREFS

Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303546, A303707, A303708, A316983, A320800-A320810.

Sequence in context: A007718 A297972 A275057 * A089264 A121399 A212421

Adjacent sequences:  A320804 A320805 A320806 * A320808 A320809 A320810

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 07 2018

STATUS

approved

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Last modified June 13 11:32 EDT 2021. Contains 344990 sequences. (Running on oeis4.)