A191970 -id:A191970 - 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

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

A321270 Number of connected multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 5, 4, 7, 3, 11, 7, 10, 1, 15, 9, 22, 7, 19, 12, 30, 5, 22, 19, 28, 14, 42, 22, 56, 1, 33, 30, 42, 20, 77, 45

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 a(2) = 1 through a(12) = 3 connected 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}}          {{1123}}
           {{1}{122}}    {{1}{112}}    {{1}{1111}}        {{1}{123}}
           {{12}{12}}    {{11}{12}}    {{11}{111}}        {{12}{13}}
           {{2}{112}}    {{1}{1}{12}}  {{1}{1}{111}}
           {{1}{2}{12}}                {{1}{11}{11}}
                                       {{1}{1}{1}{11}}
                                       {{1}{1}{1}{1}{1}}
The a(18) = 9, a(27) = 28, and a(36) = 20 connected multiset partitions of {1,1,2,2,3}, {1,1,2,2,3,3}, and {1,1,2,2,3,4} respectively:
  {{1,1,2,2,3}}      {{1,1,2,2,3,3}}        {{1,1,2,2,3,4}}
  {{1},{1,2,2,3}}    {{1},{1,2,2,3,3}}      {{1},{1,2,2,3,4}}
  {{1,2},{1,2,3}}    {{1,1,2},{2,3,3}}      {{1,1,2},{2,3,4}}
  {{1,3},{1,2,2}}    {{1,1,3},{2,2,3}}      {{1,2},{1,2,3,4}}
  {{2},{1,1,2,3}}    {{1,2},{1,2,3,3}}      {{1,2,2},{1,3,4}}
  {{2,3},{1,1,2}}    {{1,2,2},{1,3,3}}      {{1,2,3},{1,2,4}}
  {{1},{1,2},{2,3}}  {{1,2,3},{1,2,3}}      {{1,3},{1,2,2,4}}
  {{1},{2},{1,2,3}}  {{1,3},{1,2,2,3}}      {{1,4},{1,2,2,3}}
  {{2},{1,2},{1,3}}  {{2},{1,1,2,3,3}}      {{2},{1,1,2,3,4}}
                     {{2,3},{1,1,2,3}}      {{2,3},{1,1,2,4}}
                     {{3},{1,1,2,2,3}}      {{2,4},{1,1,2,3}}
                     {{1},{1,2},{2,3,3}}    {{1},{1,2},{2,3,4}}
                     {{1},{1,3},{2,2,3}}    {{1},{2},{1,2,3,4}}
                     {{1},{2},{1,2,3,3}}    {{1,2},{1,3},{2,4}}
                     {{1,2},{1,3},{2,3}}    {{1,2},{1,4},{2,3}}
                     {{1},{2,3},{1,2,3}}    {{1},{2,3},{1,2,4}}
                     {{1},{3},{1,2,2,3}}    {{1},{2,4},{1,2,3}}
                     {{2},{1,2},{1,3,3}}    {{2},{1,2},{1,3,4}}
                     {{2},{1,3},{1,2,3}}    {{2},{1,3},{1,2,4}}
                     {{2},{2,3},{1,1,3}}    {{2},{1,4},{1,2,3}}
                     {{2},{3},{1,1,2,3}}
                     {{3},{1,2},{1,2,3}}
                     {{3},{1,3},{1,2,2}}
                     {{3},{2,3},{1,1,2}}
                     {{1},{2},{1,3},{2,3}}
                     {{1},{2},{3},{1,2,3}}
                     {{1},{3},{1,2},{2,3}}
                     {{2},{3},{1,2},{1,3}}

Cf. A007718, A007719, A056156, A181821, A191970, A300913, A305193, A305936, A318284, A318286, A319557, A321272.

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.

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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.

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

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A321270 Number of connected multiset partitions of a multiset whose multiplicities are the prime indices of n.

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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.

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