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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 1-3 of 3 results.

A320665 Number of non-isomorphic multiset partitions of weight n with no singletons or vertices that appear only once.

Original entry on oeis.org

1, 0, 1, 1, 5, 6, 27, 47, 169, 406, 1327, 3790, 12560, 39919, 136821, 470589, 1687981, 6162696, 23173374, 88981796, 349969596, 1405386733, 5764142220, 24111709328, 102825231702, 446665313598, 1975339030948, 8888051121242, 40667889052853, 189126710033882, 893526261542899
Offset: 0

Views

Author

Gus Wiseman, Oct 18 2018

Keywords

Comments

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}}. This sequence counts non-isomorphic multiset partitions with no singletons whose dual also has no singletons.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 27 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}    {{1,1,1,1,1,1}}
                      {{1,1,2,2}}    {{1,1,2,2,2}}    {{1,1,1,2,2,2}}
                      {{1,1},{1,1}}  {{1,1},{1,1,1}}  {{1,1,2,2,2,2}}
                      {{1,1},{2,2}}  {{1,1},{1,2,2}}  {{1,1,2,2,3,3}}
                      {{1,2},{1,2}}  {{1,1},{2,2,2}}  {{1,1},{1,1,1,1}}
                                     {{1,2},{1,2,2}}  {{1,1,1},{1,1,1}}
                                                      {{1,1},{1,2,2,2}}
                                                      {{1,1,1},{2,2,2}}
                                                      {{1,1,2},{1,2,2}}
                                                      {{1,1},{2,2,2,2}}
                                                      {{1,1,2},{2,2,2}}
                                                      {{1,1},{2,2,3,3}}
                                                      {{1,1,2},{2,3,3}}
                                                      {{1,2},{1,1,2,2}}
                                                      {{1,2},{1,2,2,2}}
                                                      {{1,2},{1,2,3,3}}
                                                      {{1,2,2},{1,2,2}}
                                                      {{1,2,3},{1,2,3}}
                                                      {{2,2},{1,1,2,2}}
                                                      {{1,1},{1,1},{1,1}}
                                                      {{1,1},{1,2},{2,2}}
                                                      {{1,1},{2,2},{2,2}}
                                                      {{1,1},{2,2},{3,3}}
                                                      {{1,1},{2,3},{2,3}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,2},{2,2}}
                                                      {{1,2},{1,3},{2,3}}

Links

Crossrefs

Row sums of A369287.
Cf. A001055, A007716, A306005, A317794, A318871, A320663, A320664, A321760.

Programs

  • PARI
    \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A-x*sv(1)), sExp(A-x*sv(1))))} \\ Andrew Howroyd, Jan 17 2023
  • PARI
    Vec(G(20,1)) \\ G defined in A369287. - Andrew Howroyd, Jan 28 2024

Extensions

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

A369286 Triangle read by rows: T(n,k) is the number of non-isomorphic multiset partitions of weight n with k parts and no constant parts or vertices that appear in only one part, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 5, 2, 0, 0, 6, 3, 0, 0, 16, 16, 5, 0, 0, 22, 44, 13, 0, 0, 45, 135, 82, 11, 0, 0, 64, 338, 301, 52, 0, 0, 119, 880, 1233, 382, 34, 0, 0, 171, 2024, 4090, 1936, 211, 0, 0, 294, 4674, 13474, 9500, 1843, 87, 0, 0, 433, 10191, 40532, 40817, 11778, 873
Offset: 0

Views

Author

Andrew Howroyd, Jan 28 2024

Keywords

Comments

T(n,k) is the number of nonnegative integer matrices with sum of values n, k rows and every row and column having at least two nonzero entries up to permutation of rows and columns.

Examples

			Triangle begins:
  1;
  0;
  0, 0;
  0, 0;
  0, 0,   1;
  0, 0,   1;
  0, 0,   5,    2;
  0, 0,   6,    3;
  0, 0,  16,   16,    5;
  0, 0,  22,   44,   13;
  0, 0,  45,  135,   82,   11;
  0, 0,  64,  338,  301,   52;
  0, 0, 119,  880, 1233,  382,  34;
  0, 0, 171, 2024, 4090, 1936, 211;
  ...
The T(6,2) = 5 multiset partitions are:
  {{1,1,1,2}, {1,2}},
  {{1,1,2,2}, {1,2}},
  {{1,1,2}, {1,1,2}},
  {{1,1,2}, {1,2,2}},
  {{1,2,3}, {1,2,3}}.
The corresponding T(6,2) = 5 matrices are:
  [3 1]  [2 2]  [2 1]  [2 1]  [1 1 1]
  [1 1]  [1 1]  [2 1]  [1 2]  [1 1 1]
The T(6,3) = 2 matrices are:
  [1 1]  [1 1 0]
  [1 1]  [1 0 1]
  [1 1]  [0 1 1]

Links

Crossrefs

Row sums are A321760.
Cf. A307316, A369287.

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))}
H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); eta(x + O(x*x^k))*(1 + x*Ser(K(q,t,k))) + x*(1-c)/(1-x) - 1}
G(n,y=1)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, subst(H(q, t, n\t)*y^t/t, x, x^t) ))); s/n!}
T(n)={my(v=Vec(G(n,'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}
{ my(A=T(15)); for(i=1, #A, print(A[i])) }

Formula

T(2*n,n) = A307316(n).

A369927 Triangle read by rows: T(n,k) is the number of non-isomorphic set multipartitions (multisets of sets) of weight n with k parts and no singletons or endpoints, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 3, 5, 0, 0, 0, 3, 5, 0, 0, 1, 5, 17, 11, 0, 0, 0, 4, 20, 21, 0, 0, 1, 9, 53, 80, 34, 0, 0, 0, 6, 60, 167, 91, 0, 0, 1, 11, 121, 418, 410, 87, 0, 0, 0, 10, 149, 816, 1189, 402
Offset: 0

Views

Author

Andrew Howroyd, Feb 06 2024

Keywords

Comments

A singleton is a part of size 1. An endpoint is a vertex that appears in only one part.
T(n,k) is the number of binary matrices with n 1's, k rows and every row and column sum at least two up to permutation of rows and columns.

Examples

			Triangle begins:
  1;
  0;
  0, 0;
  0, 0;
  0, 0, 1;
  0, 0, 0;
  0, 0, 1,  2;
  0, 0, 0,  1;
  0, 0, 1,  3,   5;
  0, 0, 0,  3,   5;
  0, 0, 1,  5,  17,  11;
  0, 0, 0,  4,  20,  21;
  0, 0, 1,  9,  53,  80,   34;
  0, 0, 0,  6,  60, 167,   91;
  0, 0, 1, 11, 121, 418,  410,  87;
  0, 0, 0, 10, 149, 816, 1189, 402;
  ...
The T(4,2) = 1 partition is {{1,2},{1,2}}.
The corresponding matrix is:
   [1 1]
   [1 1]
The T(8,3) = 3 matrices are:
   [1 1 1]  [1 1 1 0]  [1 1 1 1]
   [1 1 1]  [1 1 0 1]  [1 1 0 0]
   [1 1 0]  [0 0 1 1]  [0 0 1 1]

Links

Crossrefs

Row sums are A369926.
Cf. A307316, A369286, A369287.

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)={my(g=x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g}
H(q, t, k)={my(c=sum(j=1, #q, if(t%q[j]==0, q[j]))); K(q, t, k) - c*x}
G(n, y=1)={my(s=0); forpart(q=n, s+=permcount(q)*exp(sum(t=1, n, subst(H(q, t, n\t)*y^t/t, x, x^t) ))); s/n!}
T(n)={my(v=Vec(G(n, 'y))); vector(#v, i, Vecrev(v[i], (i+1)\2))}
{ my(A=T(15)); for(i=1, #A, print(A[i])) }

Formula

T(2*n,n) = A307316(n).
Showing 1-3 of 3 results.

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