%I #9 Feb 07 2021 19:42:39
%S 1,1,2,3,7,11,29,55,155,386,1171
%N Number of non-isomorphic connected multiset partitions of weight n with no singletons that cannot be capped by a tree.
%C 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
%H Gus Wiseman, <a href="http://www.mathematica-journal.com/2017/12/every-clutter-is-a-tree-of-blobs/">Every Clutter Is a Tree of Blobs</a>, The Mathematica Journal, Vol. 19, 2017.
%e 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).
%e Non-isomorphic representatives of the a(2) = 2 through a(6) = 29 multiset partitions:
%e {{1,1}} {{1,1,1}} {{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}} {{1,1,1,2,2,2}}
%e {{1,2,3}} {{1,2,2,2}} {{1,2,2,2,2}} {{1,1,2,2,2,2}}
%e {{1,2,3,3}} {{1,2,2,3,3}} {{1,1,2,2,3,3}}
%e {{1,2,3,4}} {{1,2,3,3,3}} {{1,2,2,2,2,2}}
%e {{1,1},{1,1}} {{1,2,3,4,4}} {{1,2,2,3,3,3}}
%e {{1,2},{1,2}} {{1,2,3,4,5}} {{1,2,3,3,3,3}}
%e {{1,1},{1,1,1}} {{1,2,3,3,4,4}}
%e {{1,2},{1,2,2}} {{1,2,3,4,4,4}}
%e {{2,2},{1,2,2}} {{1,2,3,4,5,5}}
%e {{2,3},{1,2,3}} {{1,2,3,4,5,6}}
%e {{1,1},{1,1,1,1}}
%e {{1,1,1},{1,1,1}}
%e {{1,1,2},{1,2,2}}
%e {{1,2},{1,1,2,2}}
%e {{1,2},{1,2,2,2}}
%e {{1,2},{1,2,3,3}}
%e {{1,2,2},{1,2,2}}
%e {{1,2,3},{1,2,3}}
%e {{1,2,3},{2,3,3}}
%e {{1,3,4},{2,3,4}}
%e {{2,2},{1,1,2,2}}
%e {{2,2},{1,2,2,2}}
%e {{2,3},{1,2,3,3}}
%e {{3,3},{1,2,3,3}}
%e {{3,4},{1,2,3,4}}
%e {{1,1},{1,1},{1,1}}
%e {{1,2},{1,2},{1,2}}
%e {{1,2},{1,3},{2,3}}
%Y Non-isomorphic tree multiset partitions are counted by A321229, or A321231 without singletons.
%Y The version with singletons is A322110.
%Y The weak-antichain case is counted by A322138, or A322117 with singletons.
%Y Cf. A002218, A007718, A013922, A030019, A275307, A293994, A304118, A304382, A304887, A305079, A319719, A319721, A321194.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Nov 26 2018
%E Definition corrected by _Gus Wiseman_, Feb 05 2021
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