OFFSET
0,3
COMMENTS
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
The density of a multiset partition is defined to be the sum of numbers of distinct elements in each part, minus the number of parts, minus the total number of distinct elements in the whole partition. A multiset partition is a tree if it has more than one part, is connected, and has density -1. A cap is a certain kind of non-transitive coarsening of a multiset partition. For example, the four caps of {{1,1},{1,2},{2,2}} are {{1,1},{1,2},{2,2}}, {{1,1},{1,2,2}}, {{1,1,2},{2,2}}, {{1,1,2,2}}. - Gus Wiseman, Feb 05 2021
LINKS
Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.
EXAMPLE
The multiset partition C = {{1,1},{1,2,3},{2,3,3}} is not a tree but has the cap {{1,1},{1,2,3,3}} which is a tree, so C is not counted under a(8).
Non-isomorphic representatives of the a(1) = 1 through a(5) = 32 multiset partitions:
{{1}} {{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},{1}} {{1,2,3}} {{1,2,2,2}} {{1,2,2,2,2}}
{{1},{1,1}} {{1,2,3,3}} {{1,2,2,3,3}}
{{2},{1,2}} {{1,2,3,4}} {{1,2,3,3,3}}
{{1},{1},{1}} {{1},{1,1,1}} {{1,2,3,4,4}}
{{1,1},{1,1}} {{1,2,3,4,5}}
{{1},{1,2,2}} {{1},{1,1,1,1}}
{{1,2},{1,2}} {{1,1},{1,1,1}}
{{2},{1,2,2}} {{1},{1,2,2,2}}
{{3},{1,2,3}} {{1,2},{1,2,2}}
{{1},{1},{1,1}} {{2},{1,1,2,2}}
{{1},{2},{1,2}} {{2},{1,2,2,2}}
{{2},{2},{1,2}} {{2},{1,2,3,3}}
{{1},{1},{1},{1}} {{2,2},{1,2,2}}
{{2,3},{1,2,3}}
{{3},{1,2,3,3}}
{{4},{1,2,3,4}}
{{1},{1},{1,1,1}}
{{1},{1,1},{1,1}}
{{1},{1},{1,2,2}}
{{1},{2},{1,2,2}}
{{2},{1,2},{1,2}}
{{2},{1,2},{2,2}}
{{2},{2},{1,2,2}}
{{2},{3},{1,2,3}}
{{3},{1,3},{2,3}}
{{3},{3},{1,2,3}}
{{1},{1},{1},{1,1}}
{{1},{2},{2},{1,2}}
{{2},{2},{2},{1,2}}
{{1},{1},{1},{1},{1}}
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 26 2018
EXTENSIONS
Corrected by Gus Wiseman, Jan 27 2021
STATUS
approved