A322110 - OEIS

%I #16 Feb 07 2021 19:42:27

%S 1,1,3,6,15,32,86,216,628,1836,5822

%N Number of non-isomorphic connected multiset partitions of weight n that cannot be capped by a tree.

%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.

%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(1) = 1 through a(5) = 32 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},{1,2}}      {{1,1},{1,1,1}}

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

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

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

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

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

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

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

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

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

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

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

%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 Non-isomorphic tree multiset partitions are counted by A321229.

%Y The weak-antichain case is counted by A322117.

%Y The case without singletons is counted by A322118.

%Y Cf. A002218, A007718, A013922, A030019, A056156, A275307, A304118, A304382, A304887, A305052, A305079, A321194.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Nov 26 2018

%E Corrected by _Gus Wiseman_, Jan 27 2021