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.

Showing 1-7 of 7 results.

A320797 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 9, 15, 33, 60, 121
Offset: 0

Views

Author

Gus Wiseman, Nov 02 2018

Keywords

Comments

Also the number of nonnegative integer square symmetric matrices with sum of elements equal to n and no rows or columns summing to 0 or 1, up to row and column permutations.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 15 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{11}{22}}  {{11}{122}}  {{111}{222}}    {{111}{1222}}
                   {{12}{12}}  {{11}{222}}  {{112}{122}}    {{111}{2222}}
                               {{12}{122}}  {{11}{2222}}    {{112}{1222}}
                                            {{12}{1222}}    {{11}{22222}}
                                            {{22}{1122}}    {{12}{12222}}
                                            {{11}{22}{33}}  {{122}{1122}}
                                            {{11}{23}{23}}  {{22}{11222}}
                                            {{12}{13}{23}}  {{11}{12}{233}}
                                                            {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{11}{23}{233}}
                                                            {{12}{12}{333}}
                                                            {{12}{13}{233}}
                                                            {{13}{23}{123}}
Inequivalent representatives of the a(6) = 9 symmetric matrices with no rows or columns summing to 1:
  [6]
.
  [3 0]  [2 1]  [4 0]  [3 1]  [2 2]
  [0 3]  [1 2]  [0 2]  [1 1]  [2 0]
.
  [2 0 0]  [2 0 0]  [1 1 0]
  [0 2 0]  [0 1 1]  [1 0 1]
  [0 0 2]  [0 1 1]  [0 1 1]
		

Crossrefs

A321760 Number of non-isomorphic multiset partitions of weight n with no constant parts or vertices that appear in only one part.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 7, 9, 37, 79, 273, 755, 2648, 8432, 29872, 104624, 384759, 1432655, 5502563, 21533141, 86291313, 352654980, 1471073073, 6253397866, 27083003687, 119399628021, 535591458635, 2443030798539, 11326169401988, 53343974825122, 255121588496338
Offset: 0

Views

Author

Gus Wiseman, Nov 29 2018

Keywords

Comments

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n in which every row and column has at least two nonzero entries.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(4) = 1 through a(7) = 9 multiset partitions:
  {{1,2},{1,2}}  {{1,2},{1,2,2}}  {{1,1,2},{1,2,2}}    {{1,1,2},{1,2,2,2}}
                                  {{1,2},{1,1,2,2}}    {{1,2},{1,1,2,2,2}}
                                  {{1,2},{1,2,2,2}}    {{1,2},{1,2,2,2,2}}
                                  {{1,2,2},{1,2,2}}    {{1,2,2},{1,1,2,2}}
                                  {{1,2,3},{1,2,3}}    {{1,2,2},{1,2,2,2}}
                                  {{1,2},{1,2},{1,2}}  {{1,2,3},{1,2,3,3}}
                                  {{1,2},{1,3},{2,3}}  {{1,2},{1,2},{1,2,2}}
                                                       {{1,2},{1,3},{2,3,3}}
                                                       {{1,3},{2,3},{1,2,3}}
		

Crossrefs

Programs

Extensions

a(11) onwards from Andrew Howroyd, Jan 27 2024

A321407 Number of non-isomorphic multiset partitions of weight n with no constant parts.

Original entry on oeis.org

1, 0, 1, 2, 7, 13, 47, 111, 367, 1057, 3474, 11116, 38106, 131235, 470882, 1720959, 6472129, 24860957, 97779665, 392642763, 1610045000, 6732768139, 28699327441, 124600601174, 550684155992, 2476019025827, 11320106871951, 52598300581495, 248265707440448, 1189855827112636, 5787965846277749
Offset: 0

Views

Author

Gus Wiseman, Nov 29 2018

Keywords

Comments

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which every row has at least two nonzero entries.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 multiset partitions:
  {{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}}
                      {{1,2},{1,2}}  {{1,2,3,4,4}}
                      {{1,2},{3,4}}  {{1,2,3,4,5}}
                      {{1,3},{2,3}}  {{1,2},{1,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,2},{3,4,4}}
                                     {{1,2},{3,4,5}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,3},{1,2,3}}
		

Crossrefs

Programs

  • 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))}
    S(q, t, k)={sum(j=1, #q, if(t%q[j]==0, q[j]))*vector(k,i,1)}
    a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(x*Ser(K(q, t, n\t)-S(q, t, n\t))/t, x, x^t) )), n)); s/n!)} \\ Andrew Howroyd, Jan 17 2023

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 17 2023

A320799 Number of non-isomorphic (not necessarily strict) antichains of multisets of weight n with no singletons or leaves (vertices that appear only once).

Original entry on oeis.org

1, 0, 1, 1, 5, 4, 22, 27, 107, 212, 689
Offset: 0

Views

Author

Gus Wiseman, Nov 02 2018

Keywords

Comments

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

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 27 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{1122}}    {{11222}}    {{111222}}      {{1112222}}
                   {{11}{11}}  {{11}{122}}  {{112222}}      {{1122222}}
                   {{11}{22}}  {{11}{222}}  {{112233}}      {{1122333}}
                   {{12}{12}}               {{111}{111}}    {{111}{1222}}
                                            {{11}{1222}}    {{11}{12222}}
                                            {{111}{222}}    {{111}{2222}}
                                            {{112}{122}}    {{11}{12233}}
                                            {{11}{2222}}    {{111}{2233}}
                                            {{112}{222}}    {{112}{1222}}
                                            {{11}{2233}}    {{11}{22222}}
                                            {{112}{233}}    {{112}{2222}}
                                            {{122}{122}}    {{11}{22333}}
                                            {{123}{123}}    {{112}{2333}}
                                            {{11}{11}{11}}  {{113}{2233}}
                                            {{11}{12}{22}}  {{122}{1233}}
                                            {{11}{22}{22}}  {{222}{1122}}
                                            {{11}{22}{33}}  {{11}{11}{122}}
                                            {{11}{23}{23}}  {{11}{11}{222}}
                                            {{12}{12}{12}}  {{11}{12}{222}}
                                            {{12}{12}{22}}  {{11}{12}{233}}
                                            {{12}{13}{23}}  {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{12}{12}{222}}
                                                            {{12}{12}{233}}
                                                            {{12}{12}{333}}
                                                            {{12}{13}{233}}
		

Crossrefs

A321404 Number of non-isomorphic self-dual set multipartitions (multisets of sets) of weight n with no singletons.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 1, 3, 4, 6
Offset: 0

Views

Author

Gus Wiseman, Nov 15 2018

Keywords

Comments

Also the number of 0-1 symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which no row sums to 1.
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}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(4) = 1 through a(10) = 6 set multipartitions:
   4: {{1,2},{1,2}}
   6: {{1,2},{1,3},{2,3}}
   7: {{1,3},{2,3},{1,2,3}}
   8: {{2,3},{1,2,3},{1,2,3}}
   8: {{1,2},{1,2},{3,4},{3,4}}
   8: {{1,2},{1,3},{2,4},{3,4}}
   9: {{1,2,3},{1,2,3},{1,2,3}}
   9: {{1,2},{1,2},{3,4},{2,3,4}}
   9: {{1,2},{1,3},{1,4},{2,3,4}}
   9: {{1,2},{1,4},{3,4},{2,3,4}}
  10: {{1,2},{1,2},{1,3,4},{2,3,4}}
  10: {{1,2},{2,4},{1,3,4},{2,3,4}}
  10: {{1,3},{2,4},{1,3,4},{2,3,4}}
  10: {{1,4},{2,4},{3,4},{1,2,3,4}}
  10: {{1,2},{1,2},{3,4},{3,5},{4,5}}
  10: {{1,2},{1,3},{2,4},{3,5},{4,5}}
		

Crossrefs

A321402 Number of non-isomorphic strict self-dual multiset partitions of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 8, 14, 27, 53, 105
Offset: 0

Views

Author

Gus Wiseman, Nov 09 2018

Keywords

Comments

Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different and none sums to 1.
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}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 14 multiset partitions:
  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}      {{1111111}}
                   {{11}{22}}  {{11}{122}}  {{111}{222}}    {{111}{1222}}
                               {{11}{222}}  {{112}{122}}    {{111}{2222}}
                               {{12}{122}}  {{11}{2222}}    {{112}{1222}}
                                            {{12}{1222}}    {{11}{22222}}
                                            {{22}{1122}}    {{12}{12222}}
                                            {{11}{22}{33}}  {{122}{1122}}
                                            {{12}{13}{23}}  {{22}{11222}}
                                                            {{11}{12}{233}}
                                                            {{11}{22}{233}}
                                                            {{11}{22}{333}}
                                                            {{11}{23}{233}}
                                                            {{12}{13}{233}}
                                                            {{13}{23}{123}}
		

Crossrefs

A321677 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 4, 4, 16, 22, 70, 132, 375, 848, 2428, 6256, 18333, 52560, 161436, 500887, 1624969, 5384625, 18438815, 64674095, 233062429, 859831186, 3248411250, 12545820860, 49508089411, 199410275018, 819269777688, 3430680180687, 14633035575435, 63535672197070
Offset: 0

Views

Author

Gus Wiseman, Nov 16 2018

Keywords

Comments

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

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 16 set multipartitions:
  {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}    {{1,2,3,4,5,6}}
                      {{1,2},{1,2}}  {{1,2},{3,4,5}}  {{1,2,3},{1,2,3}}
                      {{1,2},{3,4}}  {{1,4},{2,3,4}}  {{1,2},{3,4,5,6}}
                      {{1,3},{2,3}}  {{2,3},{1,2,3}}  {{1,2,3},{4,5,6}}
                                                      {{1,2,5},{3,4,5}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{1,5},{2,3,4,5}}
                                                      {{3,4},{1,2,3,4}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,3},{2,3}}
                                                      {{1,2},{3,4},{3,4}}
                                                      {{1,2},{3,4},{5,6}}
                                                      {{1,2},{3,5},{4,5}}
                                                      {{1,3},{2,3},{2,3}}
                                                      {{1,3},{2,4},{3,4}}
                                                      {{1,4},{2,4},{3,4}}
		

Crossrefs

Programs

  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(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)={WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)) - Vec(sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k), -k)}
    a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, n, subst(x*Ser(K(q, t, n\t)/t),x,x^t) )); s+=permcount(q)*polcoef(exp(g), n)); s/n!)} \\ Andrew Howroyd, Jan 16 2024

Extensions

Terms a(11) and beyond from Andrew Howroyd, Sep 01 2019
Showing 1-7 of 7 results.