A069727 Number of nonisomorphic unrooted Eulerian planar maps with n edges (Eulerian means that all vertices are of even valency; there is an Eulerian cycle).

1, 1, 2, 4, 12, 34, 154, 675, 3534, 18985, 108070, 632109, 3807254, 23411290, 146734695, 934382820, 6034524474, 39457153432, 260855420489, 1741645762265, 11732357675908, 79673115468562, 545036528857605, 3753642607424647, 26010818244754788, 181266500331748878

Offset: 0

Author

Valery A. Liskovets, Apr 07 2002

Keywords

• easy
• nice
• nonn

Comments

By duality, also the number of unrooted (sensed) bipartite maps with n edges. - Andrew Howroyd, Mar 29 2021

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• M. Deryagina, On the enumeration of hypermaps which are self-equivalent with respect to reversing the colors of vertices, Preprint 2016.
• V. A. Liskovets and T. R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.

Crossrefs

Cf. A000257 (rooted), A069720, A069724, A103939 (with distinguished face), A103940 (with distinguished vertex).

Programs

Mathematica

a[n_] := (1/(2n)) * (3*2^(n-1) * Binomial[2n, n]/((n+1)*(n+2)) + Sum[ EulerPhi[n/k] * d[n/k] * 2^(k-2) * Binomial[2k, k], {k, Most[ Divisors[n]]}]) + q[n]; a[0] = 1; q[n_?EvenQ] := 2^((n-4)/2)*Binomial[ n, n/2]/(n+2); q[n_?OddQ] := 2^((n-1)/2)*Binomial[(n-1), (n-1)/2]/(n+1); d[n_] := 4-Mod[n, 2]; Table[ a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 19 2011, after Valery A. Liskovets *)

PARI

a(n) = {if(n==0, 1, sumdiv(n, d, if(dAndrew Howroyd, Mar 29 2021

Formula

a(n) = (1/(2n))*(3*2^(n-1)*binomial(2n, n)/((n+1)(n+2)) + Sum_{k=1..n-1, k|n} phi(n/k)*d(n/k)*2^(k-2)*binomial(2k, k)) + q(n) where phi is the Euler function A000010, q(n) = 2^((n-4)/2)*binomial(n, n/2)/(n+2) if n is even, q(n) = 2^((n-1)/2)*binomial(n-1, (n-1)/2)/(n+1) if n is odd, d(n)=4, if n is even and d(n)=3 if n is odd. - Valery A. Liskovets, Mar 17 2005

a(n) ~ 3 * 2^(3*n-2) / (sqrt(Pi) * n^(7/2)). - Vaclav Kotesovec, Aug 28 2019