A307316 -id:A307316 - cp's OEIS frontend

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).

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

Andrew Howroyd, Jan 28 2024

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.

			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]

Andrew Howroyd, Table of n, a(n) for n = 0..675 (rows 0..50)

Row sums are A321760.

Cf. A307316, A369287.

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])) }

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

A369290 Number of unlabeled simple graphs without endpoints with n edges.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 6, 10, 25, 68, 182, 538, 1748, 5935, 21585, 82904, 334037, 1406934, 6167455, 28033776, 131770437, 638956188, 3189940453, 16369201031, 86214798929, 465480395911, 2573390342437, 14553415319929, 84118459655982, 496514424803358, 2990633679878654

Offset: 0

Andrew Howroyd, Jan 30 2024

nonn

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

Row sums of A369932.

Cf. A004110 (n vertices), A307316 (multigraph), A342556 (connected).

\\ See also A369932 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)!)}

Euler transform of A342556.

A307317 Number of unlabeled connected leafless loopless multigraphs with n edges.

Original entry on oeis.org

1, 0, 1, 2, 4, 9, 26, 68, 217, 718, 2553, 9574, 38005, 157306, 679682, 3047699, 14150278, 67844305, 335262807, 1704500229, 8902528600, 47704608478, 261960998230, 1472618327415, 8466681788462, 49743177379613, 298407523833717, 1826531247381194, 11399711132242500, 72500116125222893, 469578870456459042

Offset: 0

Eric M. Metodiev, Apr 02 2019

nonn

Connected multigraphs with no loops and no vertices of degree 1.

The initial terms were computed with Nauty.

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

Andrew Howroyd, Table of n, a(n) for n = 0..50
P. T. Komiske, E. M. Metodiev, and J. Thaler, Cutting Multiparticle Correlators Down to Size, arXiv:1911.04491 [hep-ph], 2019-2020.
Brendan McKay and Adolfo Piperno, nauty and Traces.

Cf. A076864, A307316 (not necessarily connected), A342556 (simple graphs).

Inverse Euler transform of A307316.

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

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

Andrew Howroyd, Feb 06 2024

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.

			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]

Andrew Howroyd, Table of n, a(n) for n = 0..675 (rows 0..50)

Row sums are A369926.

Cf. A307316, A369286, A369287.

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])) }

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

A370063 Triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs without endpoints with n edges covering k vertices, 0 <= k <= n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 5, 2, 0, 0, 1, 5, 14, 10, 4, 0, 0, 1, 6, 25, 33, 18, 4, 0, 0, 1, 8, 46, 96, 90, 31, 7, 0, 0, 1, 10, 75, 227, 330, 194, 52, 8, 0, 0, 1, 12, 117, 494, 1033, 962, 416, 82, 12, 0, 0, 1, 14, 173, 982, 2847, 3908, 2591, 800, 128, 14

Offset: 0

Andrew Howroyd, Feb 08 2024

An endpoint is a vertex that appears in only one edge. Equivalently, the degree of every vertex >= 2.

			Triangle begins:
  1;
  0, 0;
  0, 0, 1;
  0, 0, 1,  1;
  0, 0, 1,  2,   2;
  0, 0, 1,  3,   5,   2;
  0, 0, 1,  5,  14,  10,    4;
  0, 0, 1,  6,  25,  33,   18,   4;
  0, 0, 1,  8,  46,  96,   90,  31,   7;
  0, 0, 1, 10,  75, 227,  330, 194,  52,  8;
  0, 0, 1, 12, 117, 494, 1033, 962, 416, 82, 12;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A307316.
Main diagonal is A002865.
Cf. A369932.

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)))}
G(n) = {my(s=O(x*x^n)); sum(k=0, n, forpart(p=k, s+=permcount(p) / edges(p, w->1-y^w+O(y*y^n)) * x^k * prod(i=1, #p, 1-(y*x)^p[i], 1+O(x^(n-k+1))) / k!)); s*(1-x)}
T(n)={my(r=Vec(substvec(G(n), [x, y], [y, x]))); vector(#r, i, Vecrev(Pol(r[i]), i)) }
{ my(A=T(10)); for(i=1, #A, print(A[i])) }

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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).

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A369290 Number of unlabeled simple graphs without endpoints with n edges.

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A307317 Number of unlabeled connected leafless loopless multigraphs with n edges.

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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).

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A370063 Triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs without endpoints with n edges covering k vertices, 0 <= k <= n.

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