A268558 Number of not necessarily connected sensed combinatorial maps with n edges.

1, 2, 8, 34, 182, 1300, 12634, 153598, 2231004, 37250236, 699699968, 14574247086, 333121322514, 8286605836248, 222824153996898, 6439779836400464, 199051769194393718, 6552226226766384216, 228826838199807593530, 8450335361750379998822, 329002470731473098130572

Offset

0,2

Comments

Original name: Arises in counting maps on a surface: see Coquereaux-Zuber (2015) for precise definition.

Number of nonisomorphic pairs (s,t) of permutations on a 2n-set where t is a fixed point free involution (i.e. all 2-cycles). Isomorphism is up to permutations of the n-set. - Andrew Howroyd, Jan 28 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 0..400 (terms 0..30 from Alois P. Heinz)

R. de Mello Koch, S. Ramgoolam, Strings from Feynman graph counting: Without large N, Phys. Rev. D 85 (2012) 026007. Different from a(7) onwards.

R. Coquereaux, J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016),

Prog

(PARI)
b(k, r)={if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
S(n, k)={sum(r=0, 2*n\k, if(k*r%2==0, x^(k*r/2)*b(k, r)), O(x*x^n))}
seq(n)={Vec(prod(k=1, 2*n, S(n, k)))} \\ Andrew Howroyd, Jan 28 2025

Crossrefs

Euler transform of A170946.

Sequence in context: A121894 A061370 A126328 * A191569 A229812 A346413

Adjacent sequences: A268555 A268556 A268557 * A268559 A268560 A268561

Keyword

nonn,changed

Author

N. J. A. Sloane, Mar 02 2016

Extensions

a(11)-a(18) from Euler transform of A170946 - R. J. Mathar, Apr 07 2022

a(0)=1 prepended and a(19)-a(20) (via A170946) from Alois P. Heinz, Jan 27 2025

Name edited by Andrew Howroyd, Jan 31 2025

Status

approved