A369290 -id:A369290 - cp's OEIS frontend

A307316 Number of unlabeled leafless loopless multigraphs with n edges.

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

Patrick T. Komiske, Apr 02 2019

nonn

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.

			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)}.

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.

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

\\ 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

Euler transform of A307317.

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

A342556 a(n) is the number of unlabeled connected graphs without endpoints with n edges.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 5, 9, 22, 60, 164, 494, 1624, 5602, 20600, 79813, 323806, 1371025, 6034105, 27513424, 129641411, 629862824, 3149541908, 16183100922, 85328280263, 461123500894, 2551342936264, 14438734591483, 83506198920054, 493163726073210, 2971858162771887

Offset: 0

Hugo Pfoertner, May 21 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50
Brendan McKay and Adolfo Piperno, Practical Graph Isomorphism, II, Journal of Symbolic Computation, 60 (2014), pp. 94-112.
Brendan McKay and Adolfo Piperno, nauty and Traces, programs for computing automorphism groups of graphs and digraphs.

Row sums of A342557.

Cf. A004108 (n vertices), A307317 (multigraph), A369290 (not necessarily connected).

Inverse Euler transform of A369290.

a(0)=1 prepended and a(25) onwards from Andrew Howroyd, Feb 01 2024

A369932 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 3, 5, 2, 0, 0, 0, 0, 2, 11, 9, 3, 0, 0, 0, 0, 1, 15, 32, 16, 4, 0, 0, 0, 0, 1, 12, 63, 76, 25, 5, 0, 0, 0, 0, 0, 8, 89, 234, 162, 39, 6, 0, 0, 0, 0, 0, 5, 97, 515, 730, 332, 60, 9

Offset: 1

Andrew Howroyd, Feb 07 2024

			Triangle begins:
  0;
  0, 0;
  0, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 1, 3,  2;
  0, 0, 0, 0, 3,  5,  2;
  0, 0, 0, 0, 2, 11,  9,  3;
  0, 0, 0, 0, 1, 15, 32, 16,  4;
  0, 0, 0, 0, 1, 12, 63, 76, 25, 5;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A369290.
Column sums are A261919.
Main diagonal is A008483.
Cf. A342557 (connected), A123551 (without endpoints).

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-1, i, Vecrev(Pol(r[i+1]/y),i)) }
{ my(A=T(12)); for(i=1, #A, print(A[i])) }

T(n,k) = A123551(k,n) - A123551(k-1,n).

cp's OEIS Frontend

A307316 Number of unlabeled leafless loopless multigraphs with n edges.

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A342556 a(n) is the number of unlabeled connected graphs without endpoints with n edges.

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A369932 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs with n edges and k vertices and without endpoints or isolated vertices.

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