A049314 The number k(GL(n,q)) of conjugacy classes in GL(n,q), q=4.

1, 3, 15, 60, 252, 1005, 4080, 16305, 65460, 261828, 1048260, 4192980, 16775955, 67103520, 268430160, 1073720415, 4294945932, 17179782540, 68719391100, 274877559420, 1099511281260, 4398045120300, 17592184654365, 70368738597600, 281474971147680

Offset

0,2

Comments

Bound: k(GL(n,q))<q^n. Asymptotics: k(GL(n,q)~q^n as n tends to infinity.

References

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. Journal, 27 (1960) 91-94.

Formula

The number a(n) of conjugacy classes in the group GL(n, q) is the coefficient of t^n in the infinite product: product k=1, 2, ... (1-t^k)/(1-qt^k) - Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001.

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(4^(k/d) - 1) ) * x^k/k). - Ilya Gutkovskiy, Sep 27 2018

Maple

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*4^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2012

Mathematica

b[n_] := Sum[EulerPhi[d]*4^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

Prog

(Magma)/* The program does not work for n>9: */ [1] cat [NumberOfClasses(GL(n, 4)) : n in [1..8]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by Vincenzo Librandi, Jan 23 2013
(PARI) x='x+O('x^30); Vec(prod(n=1, 30, (1-x^n)/(1-4*x^n))) \\ Altug Alkan, Sep 27 2018

Crossrefs

Cf. A006951, A006952, A049315, A049316.

Sequence in context: A036750 A286183 A058748 * A295505 A001655 A058749

Adjacent sequences: A049311 A049312 A049313 * A049315 A049316 A049317

Keyword

nonn

Author

Vladeta Jovovic

Status

approved