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

0,2

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.

Formula

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

Prog

(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

Crossrefs

Cf. A129115, A292206, A268558.

Sequence in context: A365879 A202365 A330620 * A371816 A175935 A389112

Adjacent sequences: A268553 A268554 A268555 * A268557 A268558 A268559

Keyword

nonn

Author

N. J. A. Sloane, Mar 02 2016

Extensions

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

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

Status

approved