A121352 Number of different, not necessarily connected, unlabeled trivalent diagrams of size n.

1, 1, 2, 4, 7, 10, 24, 37, 63, 112, 200, 318, 607, 1058, 1814, 3247, 6004, 10316, 19048, 35478, 63496, 117023, 223822, 408121, 766661, 1484363, 2775201, 5270079, 10357605, 19714259, 37970066, 75439670, 146103241, 284719527, 571706625

Offset

0,3

Comments

Equivalently, the number of isomorphism class of PSL_2(ZZ) actions on finite sets of size n.

Also the number of (r,s) pair of permutations up to simultaneous conjugation, in S_n for which r is involutive i.e. r^2 = id and s is of weak order three i.e. s^3 = id.

Links

Table of n, a(n) for n=0..34.

S. A. Vidal, Sur la Classification et le Dénombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison (in French), arXiv:0702223 [math.CO], 2006.

Maple

mu := k -> `if`( k mod 2 = 0, 2/k, 1/k ) : nu := k -> `if`( k mod 3 = 0, 3/k, 1/k ) : u := (k, n) -> add(mu(k)^(n-2*k2)/(n-2*k2)!/k2!/(2*k)^k2, k2=0..floor(n/ 2)) : v := (k, n) -> add(nu(k)^(n-3*k3)/(n-3*k3)!/k3!/(3*k)^k3, k3=0..floor(n/ 3)) : N := 100 : ZF := 1 : for k from N to 1 by -1 do ZF := rem(ZF * add(n!*k^n*u(k, n)*v(k, n)*t^(k*n), n = 0..floor(N/ k)), t^(N+1), t) ; end do : sort(ZF, t, ascending);

Mathematica

max = 34; mu[k_] := If[Mod[k, 2] == 0, 2/k, 1/k]; nu[k_] := If[Mod[k, 3] == 0, 3/k, 1/k]; u[k_, n_] := Sum[ mu[k]^(n - 2*k2) / (((n - 2*k2)!*k2!)*(2*k)^k2), {k2, 0, Floor[n/2]}]; v[k_, n_] := Sum[ nu[k]^(n - 3*k3) / (((n - 3*k3)!*k3!)*(3*k)^k3), {k3, 0, Floor[n/3]}]; ZF = 1; For[k = max, k >= 1, k--, ZF = PolynomialMod[ ZF*Sum[ n!*k^n*u[k, n]*v[k, n]*t^(k*n), {n, 0, Floor[max/k]}], t^(max + 1)]]; CoefficientList[ZF, t](* Jean-François Alcover, Dec 05 2012, translated from Samuel Vidal's Maple program *)

Crossrefs

Unconnected version of A121350.

Unlabeled version of A121357.

Cf. also A005133, A121355, A121356.

Sequence in context: A036685 A034744 A219748 * A134126 A277340 A337490

Adjacent sequences: A121349 A121350 A121351 * A121353 A121354 A121355

Keyword

nonn

Author

Samuel A. Vidal, Jul 23 2006

Status

approved