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 1-4 of 4 results.

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}}

Links

Crossrefs

Row sums of A369286.
Cf. A001970, A007716, A050535, A055884, A120733, A317533, A320665, A320798, A320801, A320808, A321407.

Programs

Extensions

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

A307316 Number of unlabeled leafless loopless multigraphs with n edges.

Original entry on oeis.org

1, 0, 1, 2, 5, 11, 34, 87, 279, 897, 3129, 11458, 44576, 181071, 770237, 3407332, 15641159, 74270464, 364014060, 1837689540, 9540175803, 50853577811, 277976050975, 1556372791835, 8916484189284, 52220798342832, 312389223102731, 1907282708797831, 11876576923779692, 75376983176576501, 487295169002095058
Offset: 0

Views

Author

Patrick T. Komiske, Apr 02 2019

Keywords

Comments

Multigraphs with no loops and no vertices of degree 1.
The initial terms were computed with Nauty.
Conjecturally, the asymptotic number of completely symmetric polynomials of degree n up to momentum conservation in the limit as the number of particles increases.

Examples

			For n=4 the multigraphs (as sets of edges) are {(0,1),(1,2),(2,3),(3,0)}, {(0,1),(0,1),(1,2),(2,0)}, {(0,1),(0,1),(0,1),(0,1)}, {(0,1),(0,1),(1,2),(1,2)}, and {(0,1),(0,1),(2,3),(2,3)}.

Links

Crossrefs

Conjecturally the same as A226919. Possibly also A254342.
Row sums of A370063.
Cf. A050535, A307317 (connected), A369286, A369290 (simple graphs), A369927.

Programs

  • PARI
    \\ See also A370063 for a more efficient program.
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}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
seq(n)={my(s=0); forpart(p=2*n, s+=permcount(p)*prod(i=1, #p, 1-x^p[i])/edges(p, w->1-x^w + O(x*x^n))); Vec(s/(2*n)!)} \\ Andrew Howroyd, Feb 01 2024

Formula

Euler transform of A307317.

Extensions

a(0)=1 prepended and a(17) onwards from Andrew Howroyd, Feb 01 2024

A369287 Triangle read by rows: T(n,k) is the number of non-isomorphic multiset partitions of weight n with k parts and no singletons or vertices that appear only once, 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 2, 3, 0, 2, 4, 0, 4, 15, 8, 0, 4, 24, 19, 0, 7, 60, 79, 23, 0, 8, 101, 213, 84, 0, 12, 210, 615, 424, 66, 0, 14, 357, 1523, 1533, 363, 0, 21, 679, 3851, 5580, 2217, 212, 0, 24, 1142, 8963, 17836, 10379, 1575, 0, 34, 2049, 20840, 55730, 45866, 11616, 686
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 sum at least two up to permutation of rows and columns.

Examples

			Triangle begins:
  1;
  0;
  0,  1;
  0,  1;
  0,  2,    3;
  0,  2,    4;
  0,  4,   15,    8;
  0,  4,   24,   19;
  0,  7,   60,   79,    23;
  0,  8,  101,  213,    84;
  0, 12,  210,  615,   424,    66;
  0, 14,  357, 1523,  1533,   363;
  0, 21,  679, 3851,  5580,  2217,  212;
  0, 24, 1142, 8963, 17836, 10379, 1575;
  ...
The T(5,1) = 2 multiset partitions are:
   {{1,1,1,1,1}},
   {{1,1,1,2,2}}.
The corresponding T(5,1) = 2 matrices are:
   [5]  [3 2].
The T(5,2) = 4 matrices are:
   [3]  [3 0]  [2 1]  [2 1]
   [2]  [0 2]  [1 1]  [0 2],

Links

Crossrefs

Row sums are A320665.
Columns k=0..1 are A000007, A002865(n>0).
Cf. A050535, A369286.

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]))); (1 - x)*x*Ser(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) = A050535(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 1-4 of 4 results.

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