A321155 -id:A321155 - cp's OEIS frontend

A322133 Regular triangle read by rows where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with k vertices.

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 8, 3, 1, 0, 7, 17, 12, 3, 1, 0, 11, 46, 45, 18, 4, 1, 0, 15, 94, 141, 76, 23, 4, 1, 0, 22, 212, 432, 333, 124, 30, 5, 1, 0, 30, 416, 1231, 1254, 622, 178, 37, 5, 1, 0, 42, 848, 3346, 4601, 2914, 1058, 252, 45, 6, 1

Offset: 0

Gus Wiseman, Nov 27 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.

			Triangle begins:
    1
    0    1
    0    2    1
    0    3    2    1
    0    5    8    3    1
    0    7   17   12    3    1
    0   11   46   45   18    4    1
    0   15   94  141   76   23    4    1
    0   22  212  432  333  124   30    5    1
    0   30  416 1231 1254  622  178   37    5    1
    0   42  848 3346 4601 2914 1058  252   45    6    1
Non-isomorphic representatives of the multiset partitions counted in row 4:
  {{1,1,1,1}}        {{1,1,2,2}}      {{1,2,3,3}}    {{1,2,3,4}}
  {{1},{1,1,1}}      {{1,2,2,2}}      {{1,3},{2,3}}
  {{1,1},{1,1}}      {{1},{1,2,2}}    {{3},{1,2,3}}
  {{1},{1},{1,1}}    {{1,2},{1,2}}
  {{1},{1},{1},{1}}  {{1,2},{2,2}}
                     {{2},{1,2,2}}
                     {{1},{2},{1,2}}
                     {{2},{2},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A007718.

Cf. A000664, A007716, A054923, A191646, A191970, A275421, A286520, A317672, A319719, A321155, A321254, A322114.

\\ Needs G(m,n) defined in A317533 (faster PARI).
InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
T(n)={[Vecrev(p) | p <- Vec(1 + InvEulerMTS(y^n*G(n,n) + sum(k=0, n-1, y^k*(1 - y)*G(k,n))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 15 2024

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.

A322148 Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 51, 127, 125, 1, 15, 126, 574, 1347, 1296, 1, 21, 266, 1939, 8050, 17916, 16807, 1, 28, 504, 5440, 35210, 135156, 286786, 262144, 1, 36, 882, 13387, 125730, 736401, 2642122, 5368728, 4782969, 1, 45, 1452, 29854, 388190, 3239491, 17424610, 58925728, 115089813, 100000000

Offset: 0

Gus Wiseman, Nov 28 2018

			Triangle begins:
  1
  1     1
  1     3     3
  1     6    16    16
  1    10    51   127   125
  1    15   126   574  1347  1296
  1    21   266  1939  8050 17916 16807

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A322152. Last column is A000272.

Cf. A007718, A191646, A191970, A275421, A321155, A322114, A322115, A322137, A322147.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,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]]]]]]]]];
Table[If[n==0,1,Length[Select[multsubs[multsubs[Range[k],2],n],And[Union@@#==Range[k],Length[csm[#]]==1]&]]],{n,0,5},{k,1,n+1}]

Connected(v)={my(u=vector(#v)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); u}
M(n)={Mat([Col(p, -(n+1)) | p<-Connected(vector(2*n, j, 1/(1 - x + O(x*x^n) )^binomial(j+1, 2)))[1..n+1]])}
{ my(T=M(10)); for(n=1, #T, print(T[n,][1..n])) } \\ Andrew Howroyd, Nov 29 2018

Offset corrected and terms a(28) and beyond from Andrew Howroyd, Nov 29 2018

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}]

A321256 Regular triangle where T(n,k) is the number of non-isomorphic connected set systems of weight n with density -1 <= k <= n-2.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 4, 0, 0, 0, 6, 1, 0, 0, 0, 13, 5, 0, 0, 0, 0, 23, 12, 2, 0, 0, 0, 0, 49, 36, 11, 0, 0, 0, 0, 0, 100, 95, 39, 5, 0, 0, 0, 0, 0, 220, 262, 143, 32, 1, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 01 2018

A set system is a finite set of finite nonempty sets. The density of a set system is the sum of sizes of each part (weight) minus the number of parts minus the number of vertices.

			Triangle begins:
    1
    1   0
    2   0   0
    4   0   0   0
    6   1   0   0   0
   13   5   0   0   0   0
   23  12   2   0   0   0   0
   49  36  11   0   0   0   0   0
  100  95  39   5   0   0   0   0   0
  220 262 143  32   1   0   0   0   0   0

First column is A321228. Row sums are A007718.

Cf. A007716, A030019, A056156, A283877, A293607, A300913, A319557, A321155, A321253, A321254.

A321272 Number of connected multiset partitions with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 5, 1, 4, 4, 7, 3, 11, 7, 8, 1, 15, 8, 22, 7, 14, 12, 30, 5, 16, 19, 20, 14, 42, 18, 56, 1, 24, 30, 28, 18, 77, 45, 38, 14

Offset: 1

Gus Wiseman, Nov 01 2018

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

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.

			Non-isomorphic representatives of the a(2) = 1 through a(15) = 8 multiset partitions:
  {{1}}  {{11}}    {{12}}  {{111}}      {{112}}    {{1111}}
         {{1}{1}}          {{1}{11}}    {{1}{12}}  {{1}{111}}
                           {{1}{1}{1}}             {{11}{11}}
                                                   {{1}{1}{11}}
                                                   {{1}{1}{1}{1}}
.
  {{123}}  {{1122}}      {{1112}}      {{11111}}
           {{1}{122}}    {{1}{112}}    {{1}{1111}}
           {{2}{112}}    {{11}{12}}    {{11}{111}}
           {{1}{2}{12}}  {{1}{1}{12}}  {{1}{1}{111}}
                                       {{1}{11}{11}}
                                       {{1}{1}{1}{11}}
                                       {{1}{1}{1}{1}{1}}
.
  {{1123}}    {{111111}}            {{11112}}        {{11122}}
  {{1}{123}}  {{1}{11111}}          {{1}{1112}}      {{1}{1122}}
  {{12}{13}}  {{11}{1111}}          {{11}{112}}      {{11}{122}}
              {{111}{111}}          {{12}{111}}      {{2}{1112}}
              {{1}{1}{1111}}        {{1}{1}{112}}    {{1}{1}{122}}
              {{1}{11}{111}}        {{1}{11}{12}}    {{1}{2}{112}}
              {{11}{11}{11}}        {{1}{1}{1}{12}}  {{2}{11}{12}}
              {{1}{1}{1}{111}}                       {{1}{1}{2}{12}}
              {{1}{1}{11}{11}}
              {{1}{1}{1}{1}{11}}
              {{1}{1}{1}{1}{1}{1}}

Cf. A007718, A181821, A303837, A304382, A305081, A305936, A318284, A321155, A321229, A321253, A321270, A321271.

a(prime(n)) = A000041(n).

A322112 Number of non-isomorphic self-dual connected multiset partitions of weight n with no singletons and multiset density -1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 4, 4, 9, 9

Offset: 0

Gus Wiseman, Nov 26 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 dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. A multiset partition is self-dual if it is isomorphic to its dual. For example, {{1,1},{1,2,2},{2,3,3}} is self-dual, as it is isomorphic to its dual {{1,1,2},{2,2,3},{3,3}}.
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) = 1 through a(10) = 9 multiset partitions:
  {{11}}  {{111}}  {{1111}}  {{11111}}    {{111111}}    {{1111111}}
                             {{11}{122}}  {{22}{1122}}  {{111}{1222}}
                                                        {{22}{11222}}
                                                        {{11}{12}{233}}
.
  {{11111111}}      {{111111111}}        {{1111111111}}
  {{111}{11222}}    {{1111}{12222}}      {{1111}{112222}}
  {{22}{112222}}    {{22}{1122222}}      {{22}{11222222}}
  {{11}{122}{233}}  {{222}{111222}}      {{222}{1112222}}
                    {{11}{11}{12233}}    {{111}{122}{2333}}
                    {{11}{113}{2233}}    {{22}{113}{23333}}
                    {{12}{111}{2333}}    {{22}{1133}{2233}}
                    {{22}{113}{2333}}    {{33}{33}{112233}}
                    {{12}{13}{22}{344}}  {{11}{14}{223}{344}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A316983, A321155, A321255, A322111.

A322134 Regular tetrangle where T(n,k,i) is the number of unlabeled connected multiset partitions of weight n with k vertices and i edges.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 2, 1, 1, 2, 4, 2, 1, 2, 1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 1, 1, 2, 7, 6, 2, 2, 6, 4, 1, 2, 1, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 2, 1, 1, 3, 14, 17, 9, 3, 3, 17, 18, 7, 2, 9, 7, 1, 3, 1, 0, 0

Offset: 0

Gus Wiseman, Nov 27 2018

			Tetrangle begins:
  1
.
  0 0
  1
.
  0 0 0
  1 1
  1
.
  0 0 0 0
  1 1 1
  1 1
  1
.
  0 0 0 0 0
  1 2 1 1
  2 4 2
  1 2
  1
.
  0 0 0 0 0 0
  1 2 2 1 1
  2 7 6 2
  2 6 4
  1 2
  1
.
  0  0  0  0  0  0  0
  1  3  3  2  1  1
  3 14 17  9  3
  3 17 18  7
  2  9  7
  1  3
  1
.
  0  0  0  0  0  0  0  0
  1  3  4  3  2  1  1
  3 20 33 24 11  3
  4 33 59 35 10
  3 24 35 14
  2 11 10
  1  3
  1

Triangle sums are A007718.

Cf. A007716, A054923, A191646, A191970, A275421, A286520, A317533, A317672, A321155, A321254, A322114.

cp's OEIS Frontend

A322133 Regular triangle read by rows where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with k vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Examples

Crossrefs

A322148 Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A321256 Regular triangle where T(n,k) is the number of non-isomorphic connected set systems of weight n with density -1 <= k <= n-2.

Views

Author

Keywords

Comments

Examples

Crossrefs

A321272 Number of connected multiset partitions with multiset density -1, of a multiset whose multiplicities are the prime indices of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A322112 Number of non-isomorphic self-dual connected multiset partitions of weight n with no singletons and multiset density -1.

Views

Author

Keywords

Comments

Examples

Crossrefs

A322134 Regular tetrangle where T(n,k,i) is the number of unlabeled connected multiset partitions of weight n with k vertices and i edges.

Views

Author

Keywords

Examples

Crossrefs