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A121350 Number of conjugacy class of index n subgroups in PSL_2 (ZZ). 8
1, 1, 2, 2, 1, 8, 6, 7, 14, 27, 26, 80, 133, 170, 348, 765, 1002, 2176, 4682, 6931, 13740, 31085, 48652, 96682, 217152, 362779, 707590, 1597130, 2789797, 5449439, 12233848, 22245655, 43480188, 97330468, 182619250, 358968639, 800299302 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

Also the number of different connected trivalent diagrams of size n.

Also the number of (r,s) pair of permutations in S_n, up to simultaneous conjugation, which generate a transitive action and 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=1..37.

S. A. Vidal, Sur la Classification et le Dénombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison (in French), arXiv:1103.4936 [math.CO], 2006.

Qiaochu Yuan, Drawing subgroups of the modular group, Annoying Precision, Blog, November 29, 2015.

FORMULA

If A(z) = g.f. of a(n) and B(z) = g.f. of A121352 then A(z) = sum_{k > 0} mu(k)/k log(B(z^k)) (Moebius inversion formula)

MAPLE

with(numtheory, mobius) : 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 # For example. add(convert(taylor(log(add(n!*k^n*u(k, n)*v(k, n)*t^(k*n), n = 0..floor (N/k))), t=0, N+1), polynom), k=1..N) : lZF := sort (%, t, ascending) : add(mobius(k)/k*rem(subs(t=t^k, lZF), t^(N+1), t), k=1..N) : sort (%, t, ascending);

MATHEMATICA

max = 37; 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]}]; lZF[t_] = Sum[ Normal[ Series[ Log[ Sum[n!*k^n*u[k, n]*v[k, n]*t^(k*n), {n, 0, Floor[max/k]}]], {t, 0, max + 1}]], {k, 1, max}]; Rest[ CoefficientList[ Sum[ (MoebiusMu[k]*PolynomialMod[lZF[t^k], t^(max + 1)])/k, {k, 1, max}], t]] (* Jean-François Alcover, Dec 05 2012, translated from Samuel Vidal's Maple program *)

CROSSREFS

Connected version of A121352.

Unlabeled version of A121355.

Cf. also A005133, A121356, A121357.

Sequence in context: A111540 A096440 A181738 * A198569 A135080 A238182

Adjacent sequences:  A121347 A121348 A121349 * A121351 A121352 A121353

KEYWORD

nonn

AUTHOR

Samuel Alexandre Vidal (samuel.vidal(AT)free.fr), Jul 23 2006

STATUS

approved

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Last modified May 23 23:53 EDT 2017. Contains 286937 sequences.