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Number of non-isomorphic connected weight-n multiset partitions with multiset density -1.
12

%I #6 Nov 01 2018 11:37:52

%S 1,1,3,6,16,37,105,279,817,2387,7269

%N Number of non-isomorphic connected weight-n multiset partitions with multiset density -1.

%C The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

%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.

%e Non-isomorphic representatives of the a(1) = 1 through a(5) = 37 multiset partitions:

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

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

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

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

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

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

%e {{1,1},{1,1}} {{1,2,3,4,5}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

%Y Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321227, A321228, A321231.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Oct 31 2018