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A006951 Number of conjugacy classes in GL(n,2).
(Formerly M2577)
27
1, 1, 3, 6, 14, 27, 60, 117, 246, 490, 1002, 1998, 4053, 8088, 16284, 32559, 65330, 130626, 261726, 523374, 1047690, 2095314, 4192479, 8384808, 16773552, 33546736, 67101273, 134202258, 268420086, 536839446, 1073710914, 2147420250, 4294904430, 8589807438 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Unlabeled permutations of sets. - Christian G. Bower, Jan 29 2004

From Joerg Arndt, Jan 02 2013: (Start)

Set q=2 and f(m)=q^(m-1)*(q-1), then a(n) is the sum over all partitions P of n over all products prod(k=1..L, f(m_k) ) where L is the number of different parts in the partition P=[p_1^m_1, p_2^m_2, ..., p_L^m_L], see the Macdonald reference.

Setting q to a prime power gives the sequence "Number of conjugacy classes in GL(n,q)":

q=3: A006952, q=4: A049314, q=5: A049315, q=7: A049316, q=8: A182603,

q=9: A182604, q=11: A182605, q=13: A182606, q=16: A182607, q=17: A182608,

q=19: A182609, q=23: A182610, q=25: A182611, q=27: A182612.

Sequences where q is not a prime power are:

q=6: A221578, q=10: A221579, q=12: A221580,

q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

(End)

REFERENCES

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

I. G. Macdonald, Numbers of conjugacy classes in some finite classical groups, Bulletin of the Australian Mathematical Society, vol.23, no.01, pp.23-48, (February-1981).

W. D. Smith, personal communication.

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 161

N. J. A. Sloane, Transforms

FORMULA

G.f. prod(n>=1, (1-x^n)/(1-2*x^n)  ). [Joerg Arndt, Jan 02 2013]

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

Euler transform of A008965. - Christian G. Bower, Jan 29 2004

a(n) ~ 2^n - (1+sqrt(2) + (-1)^n*(1-sqrt(2))) * 2^(n/2-1). - Vaclav Kotesovec, Nov 21 2015

EXAMPLE

For the 5 partitions of 4 (namely [1^4]; [2,1^2]; [2^2]; [3,1]; [4]) we have

(f(m) = 2^(m-1)*(2-1) = 2^(m-1) and)

f([1^4]) = 2^3 = 8,

f([2,1^2]) = 1*2^1 = 2,

f([2^2]) = 2^1 = 2,

f([3,1]) = 1*1 = 1,

f([4]) = 1,

the sum is 8+2+2+1+1 = 14 = a(4).

- Joerg Arndt, Jan 02 2013

MAPLE

with(numtheory):

b:= n-> add(phi(d)*2^(n/d), d=divisors(n))/n-1:

a:= proc(n) option remember; `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..40);  # Alois P. Heinz, Oct 20 2012

MATHEMATICA

b[n_] := Sum[EulerPhi[d]*2^(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, 40}] (* Jean-Fran├žois Alcover, Feb 17 2014, after Alois P. Heinz *)

PROG

(MAGMA)/* The program does not work for n>19: */  [1] cat [NumberOfClasses(GL(n, 2)): n in [1..19]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

(PARI)

N=66; x='x+O('x^N);

gf=prod(n=1, N, (1-x^n)/(1-2*x^n)  );

v=Vec(gf)

/* Joerg Arndt, Jan 02 2013 */

CROSSREFS

Cf. A006952, A049314, A049315, A049316, A070933, A264685, A264687.

Column k=0 of A218698. - Alois P. Heinz, Nov 04 2012

Sequence in context: A282756 A030012 A001970 * A224840 A132891 A200544

Adjacent sequences:  A006948 A006949 A006950 * A006952 A006953 A006954

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Christian G. Bower, Jan 29 2004

MAGMA code edited by Vincenzo Librandi Jan 24 2013

STATUS

approved

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Last modified November 18 17:56 EST 2017. Contains 294894 sequences.