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A321390 Third Moebius transform of A007716. Number of non-isomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods and whose dual is also an aperiodic multiset partition.

%I A321390 #5 Nov 09 2018 07:56:43
%S A321390 1,1,1,7,24,88,265,907,2929,9918,33931,119366,428314,1574221,5913415,
%T A321390 22699536,88994103,356058537,1453049451,6044132791,25612496016,
%U A321390 110503624870,485160989937,2166488899639,9835208617114,45370059225048
%N A321390 Third Moebius transform of A007716. Number of non-isomorphic aperiodic multiset partitions of weight n whose parts have relatively prime periods and whose dual is also an aperiodic multiset partition.
%C A321390 The Moebius transform c of a sequence b is c(n) = Sum_{d|n} mu(d) * b(n/d).
%C A321390 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 and the nonzero entries are relatively prime, up to row and column permutations.
%C A321390 A multiset is aperiodic if its multiplicities are relatively prime. The period of a multiset is the GCD of its multiplicities.
%C A321390 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 A321390 The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%e A321390 Non-isomorphic representatives of the a(1) = 1 through a(4) = 24 multiset partitions:
%e A321390   {{1}}  {{1},{2}}  {{1,2,2}}      {{1,2,2,2}}
%e A321390                     {{1},{1,1}}    {{1,2,3,3}}
%e A321390                     {{1},{2,2}}    {{1},{1,1,1}}
%e A321390                     {{1},{2,3}}    {{1},{1,2,2}}
%e A321390                     {{2},{1,2}}    {{1},{2,2,2}}
%e A321390                     {{1},{2},{2}}  {{1,2},{2,2}}
%e A321390                     {{1},{2},{3}}  {{1},{2,3,3}}
%e A321390                                    {{1,2},{3,3}}
%e A321390                                    {{1},{2,3,4}}
%e A321390                                    {{1,3},{2,3}}
%e A321390                                    {{2},{1,2,2}}
%e A321390                                    {{3},{1,2,3}}
%e A321390                                    {{1},{1},{1,1}}
%e A321390                                    {{1},{1},{2,2}}
%e A321390                                    {{1},{1},{2,3}}
%e A321390                                    {{1},{2},{1,2}}
%e A321390                                    {{1},{2},{2,2}}
%e A321390                                    {{1},{2},{3,3}}
%e A321390                                    {{1},{2},{3,4}}
%e A321390                                    {{1},{3},{2,3}}
%e A321390                                    {{2},{2},{1,2}}
%e A321390                                    {{1},{2},{2},{2}}
%e A321390                                    {{1},{2},{3},{3}}
%e A321390                                    {{1},{2},{3},{4}}
%Y A321390 Cf. A000740, A000837, A007716, A007916, A100953, A301700, A303386, A303431, A303546, A303547, A316983, A320800-A320810.
%K A321390 nonn
%O A321390 0,4
%A A321390 _Gus Wiseman_, Nov 08 2018