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A321412 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and with aperiodic parts. 1
1, 0, 0, 0, 1, 1, 3, 4, 12, 20, 42 (list; graph; refs; listen; history; text; internal format)



A multiset is aperiodic if its multiplicities are relatively prime.

Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with no row or column having a common divisor > 1 or summing to 1.

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}}.

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


Table of n, a(n) for n=0..10.


Non-isomorphic representatives of the a(5) = 1 through a(8) = 12 multiset partitions:

{{12}{12}}  {{12}{122}}  {{112}{122}}    {{112}{1222}}    {{1112}{1222}}

                         {{12}{1222}}    {{12}{12222}}    {{112}{12222}}

                         {{12}{13}{23}}  {{12}{13}{233}}  {{12}{122222}}

                                         {{13}{23}{123}}  {{122}{11222}}










Cf. A000219, A007716, A120733, A138178, A302545, A316983, A319616.

Cf. A320796, A320797, A320803, A320806, A320809, A320813, A321408, A321410, A321411.

Sequence in context: A124817 A306005 A124818 * A124638 A124639 A279696

Adjacent sequences:  A321409 A321410 A321411 * A321413 A321414 A321415




Gus Wiseman, Nov 16 2018



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Last modified November 30 18:35 EST 2021. Contains 349424 sequences. (Running on oeis4.)