A006951 -id:A006951 - cp's OEIS frontend

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

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V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.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

A336127 Number of ways to split a composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 2, 8, 16, 48, 144, 352, 896, 2432, 7168, 16896, 46080, 114688, 303104, 843776, 2080768, 5308416, 13762560, 34865152, 87818240, 241172480, 583008256, 1503657984, 3762290688, 9604956160, 23689428992, 60532195328, 156397207552, 385137770496, 967978254336

Offset: 0

Gus Wiseman, Jul 09 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(4) = 16 splits:
  ()  (1)  (2)    (3)        (4)
           (1,1)  (1,2)      (1,3)
                  (2,1)      (2,2)
                  (1,1,1)    (3,1)
                  (1),(2)    (1,1,2)
                  (2),(1)    (1,2,1)
                  (1),(1,1)  (1),(3)
                  (1,1),(1)  (2,1,1)
                             (3),(1)
                             (1,1,1,1)
                             (1),(1,2)
                             (1),(2,1)
                             (1,2),(1)
                             (2,1),(1)
                             (1),(1,1,1)
                             (1,1,1),(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A074854.
Starting with a strict composition gives A336128.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Compositions of partitions are A323583.
Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336130, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]

a(n) = Sum_{k=0..n} 2^(n-k) k! A008289(n,k).

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

A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

Original entry on oeis.org

1, 1, 1, 5, 5, 9, 29, 37, 57, 89, 265, 309, 521, 745, 1129, 3005, 3545, 5685, 8201, 12265, 16629, 41369, 48109, 77265, 107645, 160681, 214861, 316913, 644837, 798861, 1207445, 1694269, 2437689, 3326705, 4710397, 6270513, 12246521, 14853625, 22244569, 30308033, 43706705, 57926577, 82166105, 107873221, 148081785, 257989961, 320873065, 458994657, 628016225, 875485585, 1165065733

Offset: 0

Gus Wiseman, Jul 10 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(5) = 5 splits:
  ()  (1)  (2)  (3)     (4)     (5)
                (12)    (13)    (14)
                (21)    (31)    (23)
                (1)(2)  (1)(3)  (32)
                (2)(1)  (3)(1)  (41)
                                (1)(4)
                                (2)(3)
                                (3)(2)
                                (4)(1)
The a(6) = 29 splits:
  (6)    (1)(5)   (1)(2)(3)
  (15)   (2)(4)   (1)(3)(2)
  (24)   (4)(2)   (2)(1)(3)
  (42)   (5)(1)   (2)(3)(1)
  (51)   (1)(23)  (3)(1)(2)
  (123)  (1)(32)  (3)(2)(1)
  (132)  (13)(2)
  (213)  (2)(13)
  (231)  (2)(31)
  (312)  (23)(1)
  (321)  (31)(2)
         (32)(1)

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

The version with equal instead of different sums is A336130.
Starting with a non-strict composition gives A336127.
Starting with a partition gives A336131.
Starting with a strict partition gives A336132.
Partitions of partitions are A001970.
Partitions of compositions are A075900.
Compositions of compositions are A133494.
Set partitions with distinct block-sums are A275780.
Compositions of partitions are A323583.
Cf. A006951, A063834, A271619, A279375, A305551, A326519, A317508, A318684, A336133, A336134, A336135.

splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]

a(31)-a(50) from Max Alekseyev, Feb 14 2024

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

A182609 Number of conjugacy classes in GL(n,19).

Original entry on oeis.org

1, 18, 360, 6840, 130302, 2475720, 47045520, 893864520, 16983555840, 322687560618, 6131066120640, 116490256285320, 2213314916460120, 42052983412605480, 799006685733239040, 15181127028931412160, 288441413566677788022, 5480386857766875373560

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, A182608, A182610, A182611, A182612.

/* The program does not work for n>4: */ [1] cat [NumberOfClasses(GL(n, 19)) : n in [1..4]];

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*19^(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]*19^(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)/(1-19*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 24 2013 */

G.f.: Product_{k>=1} (1-x^k)/(1-19*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

cp's OEIS Frontend

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

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A336127 Number of ways to split a composition of n into contiguous subsequences with different sums.

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

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

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A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

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

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Magma

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Mathematica

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

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