A268556 Number of pairs (tau, sigma) of permutations of a set of size 4*n, where tau (resp. sigma) has only 2-cycles (resp. 4-cycles), up to simultaneous conjugacy.

1, 2, 10, 54, 491, 6430, 119475, 2775582, 76733201, 2439149685, 87453344290, 3488115999471, 153144951882415, 7338420391031823, 381071098250317995, 21315652618569993733, 1277715228291442258979, 81707184260073101216920, 5552193525061715345715130, 399514236526927579390940395

Offset: 0

Author

N. J. A. Sloane, Mar 02 2016

Keywords

• nonn

Comments

a(n) is the number of not necessarily connected 4-regular sensed combinatorial maps on an orientable surface with n vertices (and therefore 2n edges). - Andrew Howroyd, Jan 29 2025

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..300
• Robert Coquereaux and Jean-Bernard Zuber, Maps, immersions and permutations, J. Knot Theory Ramifications 25, 1650047 (2016); arXiv:1507.03163 [math.CO], 2015-2016.

Crossrefs

Cf. A129115, A292206, A268558.

Programs

PARI

D(m,k)={my(g=gcd(m,k)); sumdiv(g, d, my(j=m/d); x^j*eulerphi(d)*k^(j-1)/j)}
seq(n)={my(m=4,t=m*n); Vec(prod(k=1, t, my(A=O(x^(t\k+1)), p=serconvol(exp(A + D(m,k)), exp(A + D(2,k)))); sum(r=0, t\k, if(k*r%m==0, r!*polcoef(p,r)/(k^r)*x^(k*r/m)), O(x*x^n)) ))} \\ Andrew Howroyd, Jan 29 2025

Formula

Euler transform of A292206. - Andrey Zabolotskiy, Jan 14 2025

Extensions

a(0) and terms a(10)-a(17) from Andrey Zabolotskiy, Jan 23 2025

a(18) onwards from Andrew Howroyd, Jan 27 2025