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A049316
The number k(GL(n,q)) of conjugacy classes in GL(n,q), q=7.
24
1, 6, 48, 336, 2394, 16752, 117600, 823152, 5764416, 40350870, 282472512, 1977307248, 13841268048, 96888873648, 678222936384, 4747560552384, 33232929612330, 232630507267536, 1628413591207536, 11398895138319024, 79792266250574640, 558545863753891104
OFFSET
0,2
REFERENCES
V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
LINKS
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*(7^(k/d) - 1) ) * x^k/k). - Ilya Gutkovskiy, Sep 27 2018
MAPLE
with(numtheory):
b:= proc(n) b(n):= add(phi(d)*7^(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]*7^(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>8: */ [1] cat [NumberOfClasses(GL(n, 7)): 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)/(1-7*x^n))) \\ Altug Alkan, Sep 27 2018
CROSSREFS
KEYWORD
nonn
STATUS
approved