A006951 Number of conjugacy classes in GL(n,2). (Formerly M2577)

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

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 Product_{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)

From Gus Wiseman, Jan 21 2019: (Start)

Also the number of ways to split an integer partition of n into consecutive constant subsequences. For example, the a(5) = 27 ways (subsequences shown as rows) are:

5 11111

.

4 3 3 22 2 1111 1 111 11

1 2 11 1 111 1 1111 11 111

.

3 2 2 2 111 1 1 11 11 1

1 2 11 1 1 111 1 11 1 11

1 1 1 11 1 1 111 1 11 11

.

2 11 1 1 1

1 1 11 1 1

1 1 1 11 1

1 1 1 1 11

.

1

1

1

1

1

(End)

References

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

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 161

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).

N. J. A. Sloane, Transforms

Formula

G.f.: Product_{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 Product_{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

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(2^(k/d) - 1) ) * x^k/k). - Ilya Gutkovskiy, Sep 27 2018

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-François Alcover, Feb 17 2014, after Alois P. Heinz *)
Table[Sum[2^(Length[ptn]-Length[Split[ptn]]), {ptn, IntegerPartitions[n]}], {n, 30}] (* Gus Wiseman, Jan 21 2019 *)

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; edited by Vincenzo Librandi Jan 24 2013
(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

Cf. A100471, A100883, A279784, A279786, A323433, A323582, A323583.

Sequence in context: A282756 A030012 A001970 * A224840 A356804 A345334

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

Status

approved