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, 1123396477, 2214903209

Offset: 0

Author

Samuel A. Vidal, Jul 23 2006

Keywords

• nonn

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

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• 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.

Crossrefs

Not necessarily connected version of A121350.

Unlabeled version of A121357.

Cf. also A005133, A121355, A121356.

Programs

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 *)

PARI

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)={Vec(prod(k=1, n, my(A=O(x^(n\k+1)), p=serconvol(exp(A + D(1,k) + D(3,k)), exp(A + D(1,k) + D(2,k)))); sum(r=0, n\k, r!*polcoef(p,r)/(k^r)*x^(k*r), O(x*x^n)) ))} \\ Andrew Howroyd, Jan 29 2025

Formula

Euler transform of A121350. - Andrew Howroyd, Jan 29 2025

Extensions

a(35) onwards from Andrew Howroyd, Jan 29 2025