cp's OEIS Frontend

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.

%I A268556 #18 Jan 29 2025 22:06:19
%S A268556 1,2,10,54,491,6430,119475,2775582,76733201,2439149685,87453344290,
%T A268556 3488115999471,153144951882415,7338420391031823,381071098250317995,
%U A268556 21315652618569993733,1277715228291442258979,81707184260073101216920,5552193525061715345715130,399514236526927579390940395
%N 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.
%C A268556 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
%H A268556 Andrew Howroyd, <a href="/A268556/b268556.txt">Table of n, a(n) for n = 0..300</a>
%H A268556 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://doi.org/10.1142/S0218216516500474">Maps, immersions and permutations</a>, J. Knot Theory Ramifications 25, 1650047 (2016); arXiv:<a href="https://arxiv.org/abs/1507.03163">1507.03163</a> [math.CO], 2015-2016.
%F A268556 Euler transform of A292206. - _Andrey Zabolotskiy_, Jan 14 2025
%o A268556 (PARI)
%o A268556 D(m,k)={my(g=gcd(m,k)); sumdiv(g, d, my(j=m/d); x^j*eulerphi(d)*k^(j-1)/j)}
%o A268556 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
%Y A268556 Cf. A129115, A292206, A268558.
%K A268556 nonn
%O A268556 0,2
%A A268556 _N. J. A. Sloane_, Mar 02 2016
%E A268556 a(0) and terms a(10)-a(17) from _Andrey Zabolotskiy_, Jan 23 2025
%E A268556 a(18) onwards from _Andrew Howroyd_, Jan 27 2025