cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A368099 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 5, 1, 0, 7, 28, 22, 5, 1, 0, 11, 66, 83, 31, 5, 1, 0, 15, 134, 252, 147, 34, 5, 1, 0, 22, 280, 726, 620, 203, 35, 5, 1, 0, 30, 536, 1946, 2283, 1069, 235, 35, 5, 1, 0, 42, 1043, 4982, 7890, 5019, 1469, 248, 35, 5, 1
Offset: 0

Views

Author

Gus Wiseman, Dec 31 2023

Keywords

Examples

			Triangle begins:
    1
    0    1
    0    2    1
    0    3    4    1
    0    5   12    5    1
    0    7   28   22    5    1
    0   11   66   83   31    5    1
    0   15  134  252  147   34    5    1
    0   22  280  726  620  203   35    5    1
    0   30  536 1946 2283 1069  235   35    5    1
    0   42 1043 4982 7890 5019 1469  248   35    5    1
    ...
Row n = 4 counts the following representatives:
  .  {{1,1,1,1}}  {{1},{1,1,1}}  {{1},{2},{1,1}}  {{1},{2},{3},{4}}
     {{1,1,1,2}}  {{1},{1,1,2}}  {{1},{2},{1,2}}
     {{1,1,2,2}}  {{1},{1,2,2}}  {{1},{2},{1,3}}
     {{1,1,2,3}}  {{1},{1,2,3}}  {{1},{2},{3,3}}
     {{1,2,3,4}}  {{1},{2,2,2}}  {{1},{2},{3,4}}
                  {{1},{2,2,3}}
                  {{1},{2,3,4}}
                  {{1,1},{1,2}}
                  {{1,1},{2,2}}
                  {{1,1},{2,3}}
                  {{1,2},{1,3}}
                  {{1,2},{3,4}}

Links

Crossrefs

Row sums are A316980, connected case A319557.
For multiset partitions we have A317533, connected A322133.
Counting connected components instead of edges gives A321194.
For normal multiset partitions we have A330787, row sums A317776.
For set multipartitions we have A334550.
For set-systems we have A368096, row-sums A283877 (connected A300913).
A000110 counts set-partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A049311 counts non-isomorphic set multipartitions, connected A056156.
A058891 counts set-systems, unlabeled A000612, connected A323818.
Cf. A255903, A296122, A302545, A306005, A317532, A317775, A317794, A317795, A319560, A368094, A368095.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

A370646 Number of non-isomorphic multiset partitions of weight n such that only one set can be obtained by choosing a different element of each block.

Original entry on oeis.org

1, 1, 2, 4, 10, 23, 62, 165, 475, 1400, 4334
Offset: 0

Views

Author

Gus Wiseman, Mar 12 2024

Keywords

Comments

A multiset partition is a finite multiset of finite nonempty multisets. The weight of a multiset partition is the sum of cardinalities of its elements.

Examples

			The multiset partition {{3},{1,3},{2,3}} has unique choice (3,1,2) so is counted under a(5).
Representatives of the a(1) = 1 through a(5) = 23 multiset partitions:
  {1}  {11}    {111}      {1111}        {11111}
       {1}{2}  {1}{22}    {1}{122}      {11}{122}
               {2}{12}    {11}{22}      {1}{1222}
               {1}{2}{3}  {12}{12}      {11}{222}
                          {1}{222}      {12}{122}
                          {12}{22}      {1}{2222}
                          {2}{122}      {12}{222}
                          {1}{2}{33}    {2}{1122}
                          {1}{3}{23}    {2}{1222}
                          {1}{2}{3}{4}  {22}{122}
                                        {1}{2}{233}
                                        {1}{22}{33}
                                        {1}{23}{23}
                                        {1}{2}{333}
                                        {1}{23}{33}
                                        {1}{3}{233}
                                        {2}{12}{33}
                                        {2}{13}{23}
                                        {2}{3}{123}
                                        {3}{13}{23}
                                        {1}{2}{3}{44}
                                        {1}{2}{4}{34}
                                        {1}{2}{3}{4}{5}

Crossrefs

For existence we have A368098, complement A368097.
Multisets of this type are ranked by A368101, see also A368100, A355529.
Subsets of this type are counted by A370584, see also A370582, A370583.
Maximal sets of this type are counted by A370585.
Partitions of this type are counted by A370594, see also A370592, A370593.
Subsets of this type are also counted by A370638, see also A370636, A370637.
Factorizations of this type are A370645, see also A368414, A368413.
Set-systems of this type are A370818, see also A367902, A367903.
A000110 counts set partitions, non-isomorphic A000041.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
Cf. A000612, A055621, A283877, A300913, A302545, A316983, A319616, A330223, A368095, A368412, A368422.
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