A322118 Number of non-isomorphic connected multiset partitions of weight n with no singletons that cannot be capped by a tree.

1, 1, 2, 3, 7, 11, 29, 55, 155, 386, 1171

Offset

0,3

Comments

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

Table of n, a(n) for n=0..10.

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(2) = 2 through a(6) = 29 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{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}}    {{1,1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}    {{1,1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}    {{1,1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}    {{1,2,2,2,2,2}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}    {{1,2,2,3,3,3}}
                      {{1,2},{1,2}}  {{1,2,3,4,5}}    {{1,2,3,3,3,3}}
                                     {{1,1},{1,1,1}}  {{1,2,3,3,4,4}}
                                     {{1,2},{1,2,2}}  {{1,2,3,4,4,4}}
                                     {{2,2},{1,2,2}}  {{1,2,3,4,5,5}}
                                     {{2,3},{1,2,3}}  {{1,2,3,4,5,6}}
                                                      {{1,1},{1,1,1,1}}
                                                      {{1,1,1},{1,1,1}}
                                                      {{1,1,2},{1,2,2}}
                                                      {{1,2},{1,1,2,2}}
                                                      {{1,2},{1,2,2,2}}
                                                      {{1,2},{1,2,3,3}}
                                                      {{1,2,2},{1,2,2}}
                                                      {{1,2,3},{1,2,3}}
                                                      {{1,2,3},{2,3,3}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{2,2},{1,1,2,2}}
                                                      {{2,2},{1,2,2,2}}
                                                      {{2,3},{1,2,3,3}}
                                                      {{3,3},{1,2,3,3}}
                                                      {{3,4},{1,2,3,4}}
                                                      {{1,1},{1,1},{1,1}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,3},{2,3}}

Crossrefs

Non-isomorphic tree multiset partitions are counted by A321229, or A321231 without singletons.

The version with singletons is A322110.

The weak-antichain case is counted by A322138, or A322117 with singletons.

Cf. A002218, A007718, A013922, A030019, A275307, A293994, A304118, A304382, A304887, A305079, A319719, A319721, A321194.

Sequence in context: A005479 A120856 A138000 * A349420 A323067 A140108

Adjacent sequences: A322115 A322116 A322117 * A322119 A322120 A322121

Keyword

nonn,more

Author

Gus Wiseman, Nov 26 2018

Extensions

Definition corrected by Gus Wiseman, Feb 05 2021

Status

approved