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A322111
Number of non-isomorphic self-dual connected multiset partitions of weight n with multiset density -1.
4
1, 1, 1, 2, 2, 5, 5, 13, 13, 37, 37
OFFSET
0,4
COMMENTS
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.
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 partition is self-dual if it is isomorphic to its dual. For example, {{1,1},{1,2,2},{2,3,3}} is self-dual, as it is isomorphic to its dual {{1,1,2},{2,2,3},{3,3}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
EXAMPLE
Non-isomorphic representatives of the a(1) = 1 through a(8) = 13 multiset partitions:
{{1}} {{1,1}}
.
{{1,1,1}} {{1,1,1,1}}
{{2},{1,2}} {{2},{1,2,2}}
.
{{1,1,1,1,1}} {{1,1,1,1,1,1}}
{{1,1},{1,2,2}} {{2},{1,2,2,2,2}}
{{2},{1,2,2,2}} {{2,2},{1,1,2,2}}
{{2},{1,3},{2,3}} {{2},{1,3},{2,3,3}}
{{3},{3},{1,2,3}} {{3},{3},{1,2,3,3}}
.
{{1,1,1,1,1,1,1}} {{1,1,1,1,1,1,1,1}}
{{1,1,1},{1,2,2,2}} {{1,1,1},{1,1,2,2,2}}
{{2},{1,2,2,2,2,2}} {{2},{1,2,2,2,2,2,2}}
{{2,2},{1,1,2,2,2}} {{2,2},{1,1,2,2,2,2}}
{{1,1},{1,2},{2,3,3}} {{1,1},{1,2,2},{2,3,3}}
{{2},{1,3},{2,3,3,3}} {{2},{1,3},{2,3,3,3,3}}
{{2},{2,2},{1,2,3,3}} {{2},{1,3,3},{2,2,3,3}}
{{3},{1,2,2},{2,3,3}} {{3},{3},{1,2,3,3,3,3}}
{{3},{3},{1,2,3,3,3}} {{3},{3,3},{1,2,2,3,3}}
{{1},{1},{1,4},{2,3,4}} {{2},{1,3},{2,4},{3,4,4}}
{{2},{1,3},{2,4},{3,4}} {{3},{3},{1,2,4},{3,4,4}}
{{3},{4},{1,4},{2,3,4}} {{3},{4},{1,4},{2,3,4,4}}
{{4},{4},{4},{1,2,3,4}} {{4},{4},{4},{1,2,3,4,4}}
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 26 2018
STATUS
approved