A003606 a(n) = number of types of conjugacy classes in GL(n,q) (this is independent of q).
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
Offset: 1
Examples
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 non-diagonalizable 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).
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), 402-447.
- 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), 274-282.
Programs
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GAP
a := function(n) local k,sum; sum := 0; for k in [0..n-1] do sum := sum + a(k)*g(n-k); 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
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GAP
a := function(n) if( n = 0) then return 1; else return Sum([0..n], i -> t(i) * Sum(DivisorsInt(n-i), d -> d * NrPartitions(d) * Sigma(n/d)) )/n; fi; end;; # Brett Witty (witty(AT)maths.anu.edu.au), Jul 12 2006
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Mathematica
m = 32; f[x_] = Product[1/(1-x^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] (* Jean-François Alcover, Aug 01 2011, after g.f. *)
Formula
G.f.: Product_{k >= 1} f(x^k)^p_k, where f(x) = Product_{k >= 0} 1/(1-x^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(n-k+1) where g(n) = Sum_{i*j | 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
Extensions
More terms from Brett Witty (witty(AT)maths.anu.edu.au), Jul 17 2003