A320797 -id:A320797 - cp's OEIS frontend

A321403 Number of non-isomorphic self-dual set multipartitions (multisets of sets) of weight n.

1, 1, 1, 2, 4, 6, 10, 17, 32, 56, 98, 177, 335, 620, 1164, 2231, 4349, 8511, 16870, 33844, 68746, 140894, 291698, 610051, 1288594, 2745916, 5903988, 12805313, 28010036, 61764992, 137281977, 307488896, 693912297, 1577386813, 3611241900, 8324940862, 19321470086

Offset: 0

Gus Wiseman, Nov 15 2018

nonn

Also the number of symmetric (0,1)-matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns.
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.

			Non-isomorphic representatives of the a(1) = 1 through a(7) = 17 set multipartitions:
  {{1}}  {{1},{2}}  {{2},{1,2}}    {{1,2},{1,2}}      {{1},{2,3},{2,3}}
                    {{1},{2},{3}}  {{1},{1},{2,3}}    {{2},{1,3},{2,3}}
                                   {{1},{3},{2,3}}    {{3},{3},{1,2,3}}
                                   {{1},{2},{3},{4}}  {{1},{2},{2},{3,4}}
                                                      {{1},{2},{4},{3,4}}
                                                      {{1},{2},{3},{4},{5}}
.
  {{1,2},{1,3},{2,3}}        {{1,3},{2,3},{1,2,3}}
  {{3},{2,3},{1,2,3}}        {{1},{1},{1,4},{2,3,4}}
  {{1},{1},{1},{2,3,4}}      {{1},{2,3},{2,4},{3,4}}
  {{1},{2},{3,4},{3,4}}      {{1},{4},{3,4},{2,3,4}}
  {{1},{3},{2,4},{3,4}}      {{2},{1,2},{3,4},{3,4}}
  {{1},{4},{4},{2,3,4}}      {{2},{1,3},{2,4},{3,4}}
  {{2},{4},{1,2},{3,4}}      {{3},{4},{1,4},{2,3,4}}
  {{1},{2},{3},{3},{4,5}}    {{4},{4},{4},{1,2,3,4}}
  {{1},{2},{3},{5},{4,5}}    {{1},{1},{5},{2,3},{4,5}}
  {{1},{2},{3},{4},{5},{6}}  {{1},{2},{2},{2},{3,4,5}}
                             {{1},{2},{3},{4,5},{4,5}}
                             {{1},{2},{4},{3,5},{4,5}}
                             {{1},{2},{5},{5},{3,4,5}}
                             {{1},{3},{5},{2,3},{4,5}}
                             {{1},{2},{3},{4},{4},{5,6}}
                             {{1},{2},{3},{4},{6},{5,6}}
                             {{1},{2},{3},{4},{5},{6},{7}}
Inequivalent representatives of the a(6) = 10 matrices:
  [0 0 1] [1 1 0]
  [0 1 1] [1 0 1]
  [1 1 1] [0 1 1]
.
  [1 0 0 0] [1 0 0 0] [1 0 0 0] [1 0 0 0] [0 1 0 0]
  [1 0 0 0] [0 1 0 0] [0 0 1 0] [0 0 0 1] [0 0 0 1]
  [1 0 0 0] [0 0 1 1] [0 1 0 1] [0 0 0 1] [1 1 0 0]
  [0 1 1 1] [0 0 1 1] [0 0 1 1] [0 1 1 1] [0 0 1 1]
.
  [1 0 0 0 0] [1 0 0 0 0]
  [0 1 0 0 0] [0 1 0 0 0]
  [0 0 1 0 0] [0 0 1 0 0]
  [0 0 1 0 0] [0 0 0 0 1]
  [0 0 0 1 1] [0 0 0 1 1]
.
  [1 0 0 0 0 0]
  [0 1 0 0 0 0]
  [0 0 1 0 0 0]
  [0 0 0 1 0 0]
  [0 0 0 0 1 0]
  [0 0 0 0 0 1]

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

Cf. A007716, A049311, A135588, A135589, A138178, A283877, A316983, A319616.

Cf. A320796, A320797, A321403, A321404, A321405.

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}
c(p, k)={polcoef((prod(i=2, #p, prod(j=1, i-1, (1 + x^(2*lcm(p[i], p[j])) + O(x*x^k))^gcd(p[i], p[j]))) * prod(i=1, #p, my(t=p[i]); (1 + x^t + O(x*x^k))^(t%2)*(1 + x^(2*t) + O(x*x^k))^(t\2) )), k)}
a(n)={my(s=0); forpart(p=n, s+=permcount(p)*c(p, n)); s/n!} \\ Andrew Howroyd, May 31 2023

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

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

Gus Wiseman, Nov 15 2018

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.

			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}}

Cf. A007716, A049311, A135588, A138178, A283877, A302545, A316983.

Cf. A320797, A320798, A320811, A320812, A321403, A321404, A321405, A321406.

A321408 Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic.

Original entry on oeis.org

1, 1, 1, 2, 5, 9, 18, 35, 75, 153, 318

Offset: 0

Gus Wiseman, Nov 16 2018

A multiset is aperiodic if its multiplicities are relatively prime.
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 no row or column has a common divisor > 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.

			Non-isomorphic representatives of the a(1) = 1 through a(6) = 18 multiset partitions:
  {1}  {1}{2}  {2}{12}    {12}{12}      {12}{122}        {112}{122}
               {1}{2}{3}  {2}{122}      {2}{1222}        {12}{1222}
                          {1}{1}{23}    {1}{23}{23}      {2}{12222}
                          {1}{3}{23}    {1}{3}{233}      {12}{13}{23}
                          {1}{2}{3}{4}  {2}{13}{23}      {1}{23}{233}
                                        {3}{3}{123}      {1}{3}{2333}
                                        {1}{2}{2}{34}    {2}{13}{233}
                                        {1}{2}{4}{34}    {3}{23}{123}
                                        {1}{2}{3}{4}{5}  {3}{3}{1233}
                                                         {1}{1}{1}{234}
                                                         {1}{2}{34}{34}
                                                         {1}{2}{4}{344}
                                                         {1}{3}{24}{34}
                                                         {1}{4}{4}{234}
                                                         {2}{4}{12}{34}
                                                         {1}{2}{3}{3}{45}
                                                         {1}{2}{3}{5}{45}
                                                         {1}{2}{3}{4}{5}{6}

Cf. A000219, A007716, A120733, A138178, A316983, A319616.

Cf. A320796, A320797, A320803, A320804, A320805, A320806, A320807, A320809, A320813, A321410, A321411, A321412.

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

Gus Wiseman, Nov 09 2018

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.

			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}}

Cf. A000219, A007716, A059201, A302545, A316980, A316983, A319560, A319616.

Cf. A320796, A320797, A320798, A320804, A320811, A320812, A321401, A321404, A321405, A321406, A321407.

A321409 Number of non-isomorphic self-dual multiset partitions of weight n whose part sizes are relatively prime.

Original entry on oeis.org

1, 1, 1, 3, 6, 16, 27, 71, 135, 309, 621

Offset: 0

Gus Wiseman, Nov 16 2018

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, with relatively prime row sums (or column sums).
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.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions:
  {{1}}  {{1}{2}}  {{1}{22}}    {{1}{222}}      {{11}{122}}
                   {{2}{12}}    {{2}{122}}      {{11}{222}}
                   {{1}{2}{3}}  {{1}{1}{23}}    {{12}{122}}
                                {{1}{2}{33}}    {{1}{2222}}
                                {{1}{3}{23}}    {{2}{1222}}
                                {{1}{2}{3}{4}}  {{1}{22}{33}}
                                                {{1}{23}{23}}
                                                {{1}{2}{333}}
                                                {{1}{3}{233}}
                                                {{2}{12}{33}}
                                                {{2}{13}{23}}
                                                {{3}{3}{123}}
                                                {{1}{2}{2}{34}}
                                                {{1}{2}{3}{44}}
                                                {{1}{2}{4}{34}}
                                                {{1}{2}{3}{4}{5}}

Cf. A000219, A007716, A120733, A138178, A316983, A319616.

Cf. A320796, A320797, A320800, A320805, A320806, A320809, A320811, A320813, A321283, A321410, A321411, A321413.

A321413 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and relatively prime part sizes.

Original entry on oeis.org

1, 0, 0, 0, 0, 3, 0, 14, 13, 50, 65

Offset: 0

Gus Wiseman, Nov 16 2018

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, with relatively prime row sums (or column sums) and no row (or column) summing 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.

			Non-isomorphic representatives of the a(5) = 3, a(7) = 14, and a(8) = 13 multiset partitions:
  {{11}{122}}  {{111}{1222}}    {{111}{11222}}
  {{11}{222}}  {{111}{2222}}    {{111}{22222}}
  {{12}{122}}  {{112}{1222}}    {{112}{12222}}
               {{11}{22222}}    {{122}{11222}}
               {{12}{12222}}    {{11}{122}{233}}
               {{122}{1122}}    {{11}{122}{333}}
               {{22}{11222}}    {{11}{222}{333}}
               {{11}{12}{233}}  {{11}{223}{233}}
               {{11}{22}{233}}  {{12}{122}{333}}
               {{11}{22}{333}}  {{12}{123}{233}}
               {{11}{23}{233}}  {{13}{112}{233}}
               {{12}{12}{333}}  {{13}{122}{233}}
               {{12}{13}{233}}  {{23}{123}{123}}
               {{13}{23}{123}}

Cf. A000219, A007716, A120733, A138178, A302545, A316983, A319616.

Cf. A320796, A320797, A320806, A320813, A321283, A321409, A321410, A321411.

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

Gus Wiseman, Nov 16 2018

nonn

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(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}}

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

Cf. A000219, A007716, A049311, A283877, A302545, A316983, A319616.

Cf. A320797, A320798, A320804, A320811, A320812, A321404, A321406.

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

Terms a(11) and beyond from Andrew Howroyd, Sep 01 2019

A321412 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and with aperiodic parts.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 3, 4, 12, 20, 42

Offset: 0

Gus Wiseman, Nov 16 2018

A multiset is aperiodic if its multiplicities are relatively prime.
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, with no row or column having a common divisor > 1 or summing 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.

			Non-isomorphic representatives of the a(5) = 1 through a(8) = 12 multiset partitions:
{{12}{12}}  {{12}{122}}  {{112}{122}}    {{112}{1222}}    {{1112}{1222}}
                         {{12}{1222}}    {{12}{12222}}    {{112}{12222}}
                         {{12}{13}{23}}  {{12}{13}{233}}  {{12}{122222}}
                                         {{13}{23}{123}}  {{122}{11222}}
                                                          {{12}{123}{233}}
                                                          {{12}{13}{2333}}
                                                          {{13}{112}{233}}
                                                          {{13}{122}{233}}
                                                          {{13}{23}{1233}}
                                                          {{23}{123}{123}}
                                                          {{12}{12}{34}{34}}
                                                          {{12}{13}{24}{34}}

Cf. A000219, A007716, A120733, A138178, A302545, A316983, A319616.

Cf. A320796, A320797, A320803, A320806, A320809, A320813, A321408, A321410, A321411.

cp's OEIS Frontend

A321403 Number of non-isomorphic self-dual set multipartitions (multisets of sets) of weight n.

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

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A321408 Number of non-isomorphic self-dual multiset partitions of weight n whose parts are aperiodic.

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

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A321409 Number of non-isomorphic self-dual multiset partitions of weight n whose part sizes are relatively prime.

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A321413 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and relatively prime part sizes.

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

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PARI

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A321412 Number of non-isomorphic self-dual multiset partitions of weight n with no singletons and with aperiodic parts.

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