A268558 -id:A268558 - cp's OEIS frontend

A170946 Number of sensed unrooted combinatorial maps with n edges.

1, 2, 5, 20, 107, 870, 9436, 122840, 1863359, 32019826, 613981447, 12989756316, 300559406027, 7550660328494, 204687564072918, 5955893472990664, 185158932576089787, 6125200100394894738, 214837724735760642773, 7963817561236130021156, 311101285883236139915989

Offset: 0

N. J. A. Sloane, Feb 21 2010

nonn

Also number of "dessins d'enfants" with n edges. - Mark van Hoeij, Jan 23 2011

a(n) also counts the Feynman diagrams of the QED vacuum polarization with 2*n vertices: fermion lines (resp. boson lines, vertices) of the Feynman diagrams correspond to the vertices (resp. edges, darts) of the combinatorial maps, and the circular order of the edges around each vertex in a map is encoded in the topology of the corresponding Feynman diagram. - Andrey Zabolotskiy, Jan 28 2025

Andrew Howroyd, Table of n, a(n) for n = 0..400 (terms 1..30 from Antonio Breda d'Azevedo, Alexander Mednykh and Roman Nedela)
Antonio Breda d'Azevedo, Alexander Mednykh and Roman Nedela, Enumeration of maps regardless of genus: Geometric approach, Discrete Mathematics, Volume 310, 2010, Pages 1184-1203.
N. M. Adrianov, N. Ya. Amburg, V. A. Dremov, Yu. A. Levitskaya, E. M. Kreines, Yu. Yu. Kochetkov, V. F. Nasretdinova and G. B. Shabat, Catalog of dessins d'enfants with <= 4 edges, arXiv:0710.2658 [math.AG], 2007.
R. J. Mathar, Feynman diagrams of the QED vacuum polarization, vixra:1901.0148 (2019), Section V. Computed up to a(5), plotted up to a(4).
R. de Mello Koch and S. Ramgoolam, Strings from Feynman graph counting: without large N, Phys Rev D, 85 (2012) 026007; arXiv:1110.4858 [hep-th], 2011-2012. The terms in Eq. (D.10) from a(7) on are erroneous.

Row sums of A379438 and A380615.
Cf. A170947 (achiral), A214816 (unsensed).
Cf. A268558 (inv. Euler Transf.)

a(0)=1 prepended by Andrew Howroyd, Jan 28 2025

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

A170946 Number of sensed unrooted combinatorial maps with n edges.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions