

A003606


a(n) = number of types of conjugacy classes in GL(n,q) (this is independent of q).
(Formerly M3340)


2



1, 4, 8, 22, 42, 103, 199, 441, 859, 1784, 3435, 6882, 13067, 25366, 47623, 90312, 167344, 311603, 570496, 1045896, 1893886, 3426466, 6140824, 10984249, 19499214, 34526844, 60758733, 106613119, 186099976, 323883380, 561141244, 969308408
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OFFSET

1,2


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe, Table of n, a(n) for n=1..500
J. A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc., 80 (1955), 402447.
N. J. A. Sloane, Transforms
R. Steinberg, A geometric approach to the representations of the full linear group over a Galois field, Trans. Amer. Math. Soc., 71 (1951), 274282.


FORMULA

G.f.: Product_{k >= 1} f(x^k)^p_k, where f(x)=Product_{k >= 0} 1/(1x^k) = Sum_{k >= 0} p_k*x^k and p_k is the number of partitions of k (A000041).
Recurrence relation: a(n+1) = 1/(n+1) * Sum_{k=0..n} a(k)*g(nk+1) where g(n) = Sum_{ij  n} p(i)*i*j, with the sum over all ordered pairs (i, j) such that their products divide n and p(i) is the number of partitions of i. Also a(0)=1.  Brett Witty (witty(AT)maths.anu.edu.au), Jul 17 2003
Euler transform of A047968(n).  Vladeta Jovovic, Jun 23 2004
Recurrence relation: a(0)=1, a(n+1) = 1/(n+1) * Sum_{k=0..n} a(k)*g(nk+1) where g(n) = Sum_{d  n} d * A000041(d) * A000203(n/d).  Brett Witty (witty(AT)maths.anu.edu.au), Jul 12 2006


EXAMPLE

a(2) = 4 as there are four types of conjugacy classes of 2 X 2 matrices over GF(q):
* the scalar matrices (diagonal matrix with both entries the same)
* the direct sum of two scalars (diagonal matrix with both entries different)
* the nondiagonalizable Jordan block (upper triangular matrix with the same entry along the diagonal and a 1 in the superdiagonal)
* companion matrices of irreducible quadratics over GF(q)
This example can be found in Green's paper (in the references).


MATHEMATICA

m = 32; f[x_] = Product[1/(1x^k), {k, 1, m}]; gf[x_] = Product[f[x^k]^PartitionsP[k], {k, 1, m}]; Drop[ CoefficientList[ Series[gf[x], {x, 0, m}], x], 1] (* JeanFrançois Alcover, Aug 01 2011, after g.f. *)


PROG

(GAP) a := function(n) local k, sum; sum := 0; for k in [0..n1] do sum := sum + a(k)*g(nk); od; return sum/n; end;
g := function(n) local i, j, sum; for i in DivisorsInt(n) do for j in DivisorsInt(n/i) do sum := sum + NrPartitions(i)*i*j; od; od; return sum; end;;
# This code is significantly faster if you store previously computed values of a(n) and g(n).
# Brett Witty (witty(AT)maths.anu.edu.au), Jul 17 2003
(GAP) a := function(n) if( n = 0) then return 1; else return Sum([0..n], i > t(i) * Sum(DivisorsInt(ni), d > d * NrPartitions(d) * Sigma(n/d)) )/n; fi; end;; # Brett Witty (witty(AT)maths.anu.edu.au), Jul 12 2006


CROSSREFS

Cf. A001970.
Cf. A006951, A006952, A049314, A049315, A049316.
Sequence in context: A323584 A200149 A153765 * A048657 A322284 A175655
Adjacent sequences: A003603 A003604 A003605 * A003607 A003608 A003609


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane, Mira Bernstein


EXTENSIONS

More terms from Brett Witty (witty(AT)maths.anu.edu.au), Jul 17 2003


STATUS

approved



