A049315 -id:A049315 - cp's OEIS frontend

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

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

N. J. A. Sloane

nonn

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)

			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

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

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

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.

/* 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

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

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 *)

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 */

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

More terms from Christian G. Bower, Jan 29 2004

A049314 The number k(GL(n,q)) of conjugacy classes in GL(n,q), q=4.

Original entry on oeis.org

1, 3, 15, 60, 252, 1005, 4080, 16305, 65460, 261828, 1048260, 4192980, 16775955, 67103520, 268430160, 1073720415, 4294945932, 17179782540, 68719391100, 274877559420, 1099511281260, 4398045120300, 17592184654365, 70368738597600, 281474971147680

Offset: 0

Vladeta Jovovic

nonn

Bound: k(GL(n,q))

Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Alois P. Heinz, Table of n, a(n) for n = 0..500
W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. Journal, 27 (1960) 91-94.

Cf. A006951, A006952, A049315, A049316.

/* The program does not work for n>9: */ [1] cat [NumberOfClasses(GL(n,4)) : n in [1..8]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by Vincenzo Librandi, Jan 23 2013

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*4^(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

b[n_] := Sum[EulerPhi[d]*4^(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 *)

x='x+O('x^30); Vec(prod(n=1, 30, (1-x^n)/(1-4*x^n))) \\ Altug Alkan, Sep 27 2018

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*(4^(k/d) - 1) ) * x^k/k). - Ilya Gutkovskiy, Sep 27 2018

A049316 The number k(GL(n,q)) of conjugacy classes in GL(n,q), q=7.

Original entry on oeis.org

1, 6, 48, 336, 2394, 16752, 117600, 823152, 5764416, 40350870, 282472512, 1977307248, 13841268048, 96888873648, 678222936384, 4747560552384, 33232929612330, 232630507267536, 1628413591207536, 11398895138319024, 79792266250574640, 558545863753891104

Offset: 0

Vladeta Jovovic

nonn

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

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.

Cf. A006951, A006952, A049314, A049315.

/* 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

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

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 *)

x='x+O('x^30); Vec(prod(n=1, 30, (1-x^n)/(1-7*x^n))) \\ Altug Alkan, Sep 27 2018

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

A182603 Number of conjugacy classes in GL(n,8).

Original entry on oeis.org

1, 7, 63, 504, 4088, 32697, 262080, 2096577, 16776648, 134213128, 1073737224, 8589897288, 68719439943, 549755515008, 4398046212672, 35184369697407, 281474974319672, 2251799794521144, 18014398490350584, 144115187922510840, 1152921504453534648

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

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

Cf. A006951, A006952, A049314, A049315, A049316, A182604 - A182612.

/* The program does not work for n>6: */ [1] cat [NumberOfClasses(GL(n, 8)): n in [1..6]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*8^(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

b[n_] := Sum[EulerPhi[d]*8^(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, Feb 17 2014, after Alois P. Heinz *)

G.f.: prod((1-x^k)/(1-8*x^k),k=1..infinity).

Extended by D. S. McNeil, Dec 06 2010

MAGMA code edited by Vincenzo Librandi, Jan 23 2013

A182604 Number of conjugacy classes in GL(n,9).

Original entry on oeis.org

1, 8, 80, 720, 6552, 58960, 531360, 4782160, 43045920, 387413208, 3486777120, 31380993360, 282429470960, 2541865231440, 22876791858720, 205891126722080, 1853020183479912, 16677181651254480, 150094635248646000, 1350851717237225040, 12157665458621220720

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..350

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182605, A182606, A182607, A182608, A182609, A182610, A182611, A182612.

/* The program does not work for n>6: */ [1] cat [NumberOfClasses(GL(n, 9)): n in [1..6]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*9^(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

b[n_] := Sum[EulerPhi[d]*9^(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, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-9*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-9*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

MAGMA code edited by Vincenzo Librandi, Jan 24 2013

A182605 Number of conjugacy classes in GL(n,11).

Original entry on oeis.org

1, 10, 120, 1320, 14630, 160920, 1771440, 19485720, 214357440, 2357931730, 25937408640, 285311493720, 3138428201160, 34522710196920, 379749831637440, 4177248147997440, 45949729842155150, 505447028263532520, 5559917313256631160, 61159090445821012920

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182604, A182606, A182607, A182608, A182609, A182610, A182611, A182612.

N := 300;  R := PowerSeriesRing(Integers(), N);
Eltseq( &*[ (1-x^k)/(1-11*x^k) : k in [1..N] ] ); // Volker Gebhardt, Dec 07 2020

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*11^(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

b[n_] := Sum[EulerPhi[d]*11^(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, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-11*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-11*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

A182606 Number of conjugacy classes in GL(n,13).

Original entry on oeis.org

1, 12, 168, 2184, 28548, 371112, 4826640, 62746152, 815728368, 10604468628, 137858461104, 1792159992168, 23298084722808, 302875101365928, 3937376380474992, 51185892946146672, 665416609115237772, 8650415918497693704, 112455406951074120024

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182604, A182605, A182607, A182608, A182609, A182610, A182611, A182612.

/* The program does not work for n>5: */ [1] cat [NumberOfClasses(GL(n, 13)): n in [1..5]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*13^(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

b[n_] := Sum[EulerPhi[d]*13^(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, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-13*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-13*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

MAGMA code edited by Vincenzo Librandi, Jan 24 2013

A182607 Number of conjugacy classes in GL(n,16).

Original entry on oeis.org

1, 15, 255, 4080, 65520, 1048305, 16776960, 268431105, 4294962960, 68719407120, 1099511558160, 17592184926480, 281474975596815, 4503599609479680, 72057594020040960, 1152921504320590335, 18446744073423298800, 295147905174771671280, 4722366482865065107440

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182604, A182605, A182606, A182608, A182609, A182610, A182611, A182612.

/* The program does not work for n>6: */ [1] cat [NumberOfClasses(GL(n, 16)) : n in [1..6]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*16^(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

b[n_] := Sum[EulerPhi[d]*16^(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, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-16*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-16*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

MAGMA code edited by Vincenzo Librandi, Jan 24 2013

A182608 Number of conjugacy classes in GL(n,17).

Original entry on oeis.org

1, 16, 288, 4896, 83504, 1419552, 24137280, 410333472, 6975752256, 118587788080, 2015993812032, 34271894799648, 582622235726688, 9904578007265568, 168377826533765184, 2862423051073925184, 48661191875230982480, 827240261878925204256, 14063084452060314850656

Offset: 0

Klaus Brockhaus, Nov 23 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A006951, A006952, A049314, A049315, A049316, A182603, A182604, A182605, A182606, A182607, A182609, A182610, A182611, A182612.

/* The program does not work for n>4: */ [1] cat [NumberOfClasses(GL(n, 17)) : n in [1..4]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*17^(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

b[n_] := Sum[EulerPhi[d]*17^(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, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-17*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-17*x^k). - Alois P. Heinz, Nov 03 2012

More terms from Alois P. Heinz, Nov 03 2012

MAGMA code edited by Vincenzo Librandi, Jan 24 2013

Showing 1-10 of 23 results. Next

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A006952 Number of conjugacy classes in GL(n,3).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A049314 The number k(GL(n,q)) of conjugacy classes in GL(n,q), q=4.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A049316 The number k(GL(n,q)) of conjugacy classes in GL(n,q), q=7.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A182603 Number of conjugacy classes in GL(n,8).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A182604 Number of conjugacy classes in GL(n,9).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica