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

Author

Patrick T. Komiske, Apr 02 2019

Keywords

• nonn

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

• 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.

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