A321229 -id:A321229 - cp's OEIS frontend

A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

1, 3, 0, 6, 0, 0, 16, 1, 0, 0, 37, 3, 0, 0, 0, 105, 18, 2, 0, 0, 0, 279, 68, 7, 0, 0, 0, 0, 817, 293, 46, 3, 0, 0, 0, 0, 2387, 1141, 228, 17, 1, 0, 0, 0, 0, 7269, 4511, 1189, 135, 9, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 01 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

			Triangle begins:
     1
     3    0
     6    0    0
    16    1    0    0
    37    3    0    0    0
   105   18    2    0    0    0
   279   68    7    0    0    0    0
   817  293   46    3    0    0    0    0
  2387 1141  228   17    1    0    0    0    0
  7269 4511 1189  135    9    0    0    0    0    0

First column is A321229. Row sums are A007718.

Cf. A007716, A317672, A321155, A321194, A321253, A321255, A321256.

A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 35, 85, 231, 633, 1845, 5461, 16707, 51945, 164695, 529077, 1722279, 5664794, 18813369, 62996850, 212533226, 721792761, 2466135375, 8471967938, 29249059293, 101440962296, 353289339927, 1235154230060, 4333718587353, 15255879756033

Offset: 0

Christian G. Bower, Nov 15 1998

nonn

Also the number of non-isomorphic connected multigraphs with loops with n edges and multiset density -1, where the multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices. - Gus Wiseman, Nov 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1717
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Gus Wiseman, Non-isomorphic representatives of the a(1) = 2 through a(5) = 35 connected multigraphs with loops with multiset density -1.
Index entries for sequences related to trees

Essentially the same as A036251.

Cf. A000055, A007718, A007719, A038052, A191646, A303837, A321155, A321229, A321254, A321256, A322111.

max = 30; B[] = 1; Do[B[x] = x*Exp[Sum[(B[x^k] + x^k)/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; A[x_] = B[x] - B[x]^2/2 + B[x^2]/2; CoefficientList[1 + A[x] + O[x]^max, x] (* Jean-François Alcover, Jan 28 2019 *)

G.f.: B(x) - B^2(x)/2 + B(x^2)/2, where B(x) is g.f. for A036249.

A321228 Number of non-isomorphic hypertrees of weight n with singletons.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 13, 23, 49, 100, 220

Offset: 0

Gus Wiseman, Oct 31 2018

A hypertree with singletons is a connected set system (finite set of finite nonempty sets) with density -1, where the density of a set system is the sum of sizes of the parts (weight) minus the number of parts minus the number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(7) = 23 hypertrees:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}      {{1,2,3,4,5}}
                  {{2},{1,2}}  {{1,3},{2,3}}    {{1,4},{2,3,4}}
                               {{3},{1,2,3}}    {{4},{1,2,3,4}}
                               {{1},{2},{1,2}}  {{2},{1,3},{2,3}}
                                                {{2},{3},{1,2,3}}
                                                {{3},{1,3},{2,3}}
.
  {{1,2,3,4,5,6}}        {{1,2,3,4,5,6,7}}
  {{1,2,5},{3,4,5}}      {{1,2,6},{3,4,5,6}}
  {{1,5},{2,3,4,5}}      {{1,6},{2,3,4,5,6}}
  {{5},{1,2,3,4,5}}      {{6},{1,2,3,4,5,6}}
  {{1},{1,4},{2,3,4}}    {{1},{1,5},{2,3,4,5}}
  {{1,3},{2,4},{3,4}}    {{1,2},{2,5},{3,4,5}}
  {{1,4},{2,4},{3,4}}    {{1,4},{2,5},{3,4,5}}
  {{3},{1,4},{2,3,4}}    {{1,5},{2,5},{3,4,5}}
  {{3},{4},{1,2,3,4}}    {{4},{1,2,5},{3,4,5}}
  {{4},{1,4},{2,3,4}}    {{4},{1,5},{2,3,4,5}}
  {{1},{2},{1,3},{2,3}}  {{4},{5},{1,2,3,4,5}}
  {{1},{2},{3},{1,2,3}}  {{5},{1,2,5},{3,4,5}}
  {{2},{3},{1,3},{2,3}}  {{5},{1,5},{2,3,4,5}}
                         {{1},{3},{1,4},{2,3,4}}
                         {{1},{4},{1,4},{2,3,4}}
                         {{2},{1,3},{2,4},{3,4}}
                         {{2},{3},{1,4},{2,3,4}}
                         {{2},{3},{4},{1,2,3,4}}
                         {{3},{1,4},{2,4},{3,4}}
                         {{3},{4},{1,4},{2,3,4}}
                         {{4},{1,3},{2,4},{3,4}}
                         {{4},{1,4},{2,4},{3,4}}
                         {{1},{2},{3},{1,3},{2,3}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A317672, A318697, A321155, A321227, A321229.

A321253 Number of non-isomorphic strict connected weight-n multiset partitions with multiset density -1.

Original entry on oeis.org

0, 1, 2, 5, 12, 28, 78, 202, 578, 1650, 4904

Offset: 0

Gus Wiseman, Nov 01 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 28 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}}
                  {{1,2,3}}    {{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}}    {{1,2,3,4,4}}
                               {{1},{1,2,2}}    {{1,2,3,4,5}}
                               {{1,2},{2,2}}    {{1},{1,1,1,1}}
                               {{1,3},{2,3}}    {{1,1},{1,1,1}}
                               {{2},{1,2,2}}    {{1,1},{1,2,2}}
                               {{3},{1,2,3}}    {{1},{1,2,2,2}}
                               {{1},{2},{1,2}}  {{1,2},{2,2,2}}
                                                {{1,2},{2,3,3}}
                                                {{1,3},{2,3,3}}
                                                {{1,4},{2,3,4}}
                                                {{2},{1,1,2,2}}
                                                {{2},{1,2,2,2}}
                                                {{2},{1,2,3,3}}
                                                {{2,2},{1,2,2}}
                                                {{3},{1,2,3,3}}
                                                {{3,3},{1,2,3}}
                                                {{4},{1,2,3,4}}
                                                {{1},{1,2},{2,2}}
                                                {{1},{2},{1,2,2}}
                                                {{2},{1,2},{2,2}}
                                                {{2},{1,3},{2,3}}
                                                {{2},{3},{1,2,3}}
                                                {{3},{1,3},{2,3}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321194, A321228, A321229, A321254.

A322110 Number of non-isomorphic connected multiset partitions of weight n that cannot be capped by a tree.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Nov 26 2018

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

			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}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Non-isomorphic tree multiset partitions are counted by A321229.
The weak-antichain case is counted by A322117.
The case without singletons is counted by A322118.
Cf. A002218, A007718, A013922, A030019, A056156, A275307, A304118, A304382, A304887, A305052, A305079, A321194.

Corrected by Gus Wiseman, Jan 27 2021

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Nov 26 2018

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

			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}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

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.

Definition corrected by Gus Wiseman, Feb 05 2021

A321231 Number of non-isomorphic connected weight-n multiset partitions with no singletons and multiset density -1.

Original entry on oeis.org

1, 0, 2, 3, 8, 15, 42, 94, 256, 656, 1807

Offset: 0

Gus Wiseman, Oct 31 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 2 through a(5) = 15 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},{1,1}}  {{1,2,3,4,4}}
                      {{1,2},{2,2}}  {{1,2,3,4,5}}
                      {{1,3},{2,3}}  {{1,1},{1,1,1}}
                                     {{1,1},{1,2,2}}
                                     {{1,2},{2,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,2},{1,2,2}}
                                     {{3,3},{1,2,3}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A317672, A318697, A321155, A321228, A321229.

A321255 Number of connected multiset partitions with multiset density -1, of strongly normal multisets of size n, with no singletons.

Original entry on oeis.org

0, 0, 2, 3, 8, 19, 60, 183, 643, 2355, 9393

Offset: 0

Gus Wiseman, Nov 01 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing.

			The a(2) = 2 through a(5) = 19 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,1,2}}  {{1,1,1,2}}    {{1,1,1,1,2}}
           {{1,2,3}}  {{1,1,2,2}}    {{1,1,1,2,2}}
                      {{1,1,2,3}}    {{1,1,1,2,3}}
                      {{1,2,3,4}}    {{1,1,2,2,3}}
                      {{1,1},{1,1}}  {{1,1,2,3,4}}
                      {{1,1},{1,2}}  {{1,2,3,4,5}}
                      {{1,2},{1,3}}  {{1,1},{1,1,1}}
                                     {{1,1},{1,1,2}}
                                     {{1,1},{1,2,2}}
                                     {{1,1},{1,2,3}}
                                     {{1,2},{1,1,1}}
                                     {{1,2},{1,1,3}}
                                     {{1,2},{1,3,4}}
                                     {{1,3},{1,1,2}}
                                     {{1,3},{1,2,2}}
                                     {{1,3},{1,2,4}}
                                     {{1,4},{1,2,3}}
                                     {{2,3},{1,1,2}}

Cf. A000272, A030019, A052888, A134954, A304867, A317672, A321228, A321229, A321253.

A321227 Number of connected multiset partitions with multiset density -1 of strongly normal multisets of size n.

Original entry on oeis.org

0, 1, 3, 6, 17, 43, 147, 458, 1729, 6445, 27011

Offset: 0

Gus Wiseman, Oct 31 2018

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing.

			The a(1) = 1 through a(4) = 17 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
         {{1},{1}}  {{1,2,3}}      {{1,1,2,2}}
                    {{1},{1,1}}    {{1,1,2,3}}
                    {{1},{1,2}}    {{1,2,3,4}}
                    {{1},{1},{1}}  {{1},{1,1,1}}
                                   {{1,1},{1,1}}
                                   {{1},{1,1,2}}
                                   {{1,1},{1,2}}
                                   {{1},{1,2,2}}
                                   {{1},{1,2,3}}
                                   {{1,2},{1,3}}
                                   {{2},{1,1,2}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{1,2}}
                                   {{1},{2},{1,2}}
                                   {{1},{1},{1},{1}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321228, A321229, A321231.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
mensity[c_]:=Total[(Length[Union[#]]-1&)/@c]-Length[Union@@c];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Sum[Length[Select[mps[m],And[mensity[#]==-1,Length[csm[#]]==1]&]],{m,strnorm[n]}],{n,0,8}]

A321271 Number of connected factorizations of n into positive integers > 1 with z-density -1.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 7, 5, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Nov 01 2018

nonn

These are z-trees (A303837, A305081, A305253, A321279) where we relax the requirement of pairwise indivisibility.
Given a finite multiset S of positive integers greater than 1, let G(S) be the simple labeled graph with vertices the distinct elements of S and with edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. Then S is said to be connected if G(S) is a connected graph.
The z-density of a factorization S is defined to be Sum_{s in S} (omega(s) - 1) - omega(n), where omega = A001221 and n is the product of S.

			The a(72) = 8 factorizations are (2*2*3*6), (2*2*18), (2*3*12), (2*36), (3*4*6), (3*24), (4*18), (72). Missing from this list but still connected are (2*6*6),(6*12).

Cf. A001055, A001221, A030019, A286518, A303837, A304118, A304382, A305052, A305081, A305193, A305253, A319786, A321229, A321253.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[Times@@s];
Table[Length[Select[facs[n],And[zensity[#]==-1,Length[zsm[#]]==1]&]],{n,100}]

cp's OEIS Frontend

A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

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A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

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A321228 Number of non-isomorphic hypertrees of weight n with singletons.

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A321253 Number of non-isomorphic strict connected weight-n multiset partitions with multiset density -1.

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A322110 Number of non-isomorphic connected multiset partitions of weight n that cannot be capped by a tree.

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

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A321231 Number of non-isomorphic connected weight-n multiset partitions with no singletons and multiset density -1.

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A321255 Number of connected multiset partitions with multiset density -1, of strongly normal multisets of size n, with no singletons.

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A321227 Number of connected multiset partitions with multiset density -1 of strongly normal multisets of size n.

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A321271 Number of connected factorizations of n into positive integers > 1 with z-density -1.

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