A292206 -id:A292206 - cp's OEIS frontend

A380626 Array read by antidiagonals: T(n,k) is the number of sensed k-regular combinatorial maps with n vertices, n >= 0, k >= 1.

1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 2, 3, 1, 0, 1, 0, 7, 0, 1, 0, 1, 5, 29, 36, 11, 1, 0, 1, 0, 174, 0, 365, 0, 1, 0, 1, 18, 1475, 26614, 44106, 5250, 81, 1, 0, 1, 0, 16162, 0, 10107019, 0, 103801, 0, 1, 0, 1, 105, 214215, 102762168, 3703659517, 6605320523, 549530780, 2492164, 1228, 1, 0

Offset: 0

Andrew Howroyd, Jan 29 2025

The combinatorial maps considered are connected, unrooted, unlabeled, may have loops and parallel edges and are of any orientable genus.

			Array begins:
==================================================================
n\k | 1 2  3       4         5          6          7         8 ...
----+-------------------------------------------------------------
  0 | 1 1  1       1         1          1          1         1 ...
  1 | 0 1  0       2         0          5          0        18 ...
  2 | 1 1  3       7        29        174       1475     16162 ...
  3 | 0 1  0      36         0      26614          0 102762168 ...
  4 | 0 1 11     365     44106   10107019 3703659517 ...
  5 | 0 1  0    5250         0 6605320523 ...
  6 | 0 1 81  103801 549530780 ...
  7 | 0 1  0 2492164 ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Columns 2..6 (odd columns with interspersed zeros) are A000012, A129114, A292206, A380627, A380628.
Row n=1 is A007769 (with interspersed zeros).
Cf. A170946, A380622 (rooted), A380629.

InvEulerT(v)={dirdiv(Vec(log(1+x*Ser(v)),-#v), vector(#v,n,1/n))}
D(m,k)={my(g=gcd(m,k)); sumdiv(g, d, my(j=m/d); x^j*eulerphi(d)*k^(j-1)/j)}
G(n,m)={my(t=m*n); 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)) )}
T(n,k)=if(n==0, 1, InvEulerT(Vec(-1 + G(n,k), -n))[n])

A380629(n) = Sum_{d|2*n} T(d,2*n/d).

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.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Mar 02 2016

nonn

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

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.

Cf. A129115, A292206, A268558.

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

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

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

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

cp's OEIS Frontend

A380626 Array read by antidiagonals: T(n,k) is the number of sensed k-regular combinatorial maps with n vertices, n >= 0, k >= 1.

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