A320802 - OEIS

%I #14 Jun 27 2020 03:09:41

%S 1,1,2,8,26,89,274,908,2955,9926,34021,119367,428612,1574222,5914324,

%T 22699632,88997058,356058538,1453059643,6044132792,25612530061,

%U 110503625785,485161109305,2166488899640,9835209048655,45370059225137,212582814591083,1011306624492831

%N Number of non-isomorphic aperiodic multiset partitions of weight n whose dual is also an aperiodic multiset partition.

%C Also the number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns where the multiset of rows and the multiset of columns are both aperiodic, up to row and column permutations.

%C A multiset is aperiodic if its multiplicities are relatively prime.

%C 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}}.

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

%C Also the number of non-isomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods, where the period of a multiset is the GCD of its multiplicities.

%H Jinyuan Wang, <a href="/A320802/b320802.txt">Table of n, a(n) for n = 0..50</a>

%F Second Moebius transform of A007716, or Moebius transform of A303546, where the Meobius transform of a sequence b is a(n) = Sum_{d|n} mu(d) * b(n/d).

%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 26 multiset partitions:

%e   {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}

%e          {{1},{2}}  {{1,2,2}}      {{1,2,2,2}}

%e                     {{1},{1,1}}    {{1,2,3,3}}

%e                     {{1},{2,2}}    {{1},{1,1,1}}

%e                     {{1},{2,3}}    {{1},{1,2,2}}

%e                     {{2},{1,2}}    {{1,1},{2,2}}

%e                     {{1},{2},{2}}  {{1},{2,2,2}}

%e                     {{1},{2},{3}}  {{1,2},{2,2}}

%e                                    {{1},{2,3,3}}

%e                                    {{1,2},{3,3}}

%e                                    {{1},{2,3,4}}

%e                                    {{1,3},{2,3}}

%e                                    {{2},{1,2,2}}

%e                                    {{3},{1,2,3}}

%e                                    {{1},{1},{1,1}}

%e                                    {{1},{1},{2,2}}

%e                                    {{1},{1},{2,3}}

%e                                    {{1},{2},{1,2}}

%e                                    {{1},{2},{2,2}}

%e                                    {{1},{2},{3,3}}

%e                                    {{1},{2},{3,4}}

%e                                    {{1},{3},{2,3}}

%e                                    {{2},{2},{1,2}}

%e                                    {{1},{2},{2},{2}}

%e                                    {{1},{2},{3},{3}}

%e                                    {{1},{2},{3},{4}}

%Y Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A316983, A320800-A320810.

%K nonn

%O 0,3

%A _Gus Wiseman_, Nov 06 2018

%E a(26)-a(27) from _Jinyuan Wang_, Jun 27 2020