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A320812
Number of non-isomorphic aperiodic multiset partitions of weight n with no singletons.
6
1, 0, 2, 3, 10, 23, 79, 204, 670, 1974, 6521, 21003, 71944, 248055, 888565, 3240552, 12152093, 46527471, 182337383, 729405164, 2979114723, 12407307929, 52670334237, 227725915268, 1002285201807, 4487915293675, 20434064047098, 94559526594316, 444527729321513
OFFSET
0,3
COMMENTS
A multiset is aperiodic if its multiplicities are relatively prime.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
LINKS
FORMULA
a(n) = Sum_{d|n} mu(d)*A302545(n/d) for n > 0. - Andrew Howroyd, Jan 16 2023
EXAMPLE
Non-isomorphic representatives of the a(2) = 2 through a(5) = 23 multiset partitions:
{{1,1}} {{1,1,1}} {{1,1,1,1}} {{1,1,1,1,1}}
{{1,2}} {{1,2,2}} {{1,1,2,2}} {{1,1,2,2,2}}
{{1,2,3}} {{1,2,2,2}} {{1,2,2,2,2}}
{{1,2,3,3}} {{1,2,2,3,3}}
{{1,2,3,4}} {{1,2,3,3,3}}
{{1,1},{2,2}} {{1,2,3,4,4}}
{{1,2},{2,2}} {{1,2,3,4,5}}
{{1,2},{3,3}} {{1,1},{1,1,1}}
{{1,2},{3,4}} {{1,1},{1,2,2}}
{{1,3},{2,3}} {{1,1},{2,2,2}}
{{1,1},{2,3,3}}
{{1,1},{2,3,4}}
{{1,2},{1,2,2}}
{{1,2},{2,2,2}}
{{1,2},{2,3,3}}
{{1,2},{3,3,3}}
{{1,2},{3,4,4}}
{{1,2},{3,4,5}}
{{1,3},{2,3,3}}
{{1,4},{2,3,4}}
{{2,2},{1,2,2}}
{{2,3},{1,2,3}}
{{3,3},{1,2,3}}
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 08 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023
STATUS
approved