A221583 -id:A221583 - 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

A221578 A sum over partitions (q=6), see first comment.

Original entry on oeis.org

1, 5, 35, 210, 1290, 7735, 46620, 279685, 1679370, 10076190, 60464670, 362787810, 2176773305, 13060638360, 78364108620, 470184650495, 2821109573550, 16926657432510, 101559954663930, 609359727929610, 3656158427989830, 21936950567886270, 131621703769781995

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=6 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].
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:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..400

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*6^(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, Jan 24 2013

b[n_] := Sum[EulerPhi[d]*6^(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)/(1-6*x^n)  );
v=Vec(gf)

A221579 A sum over partitions (q=10), see first comment.

Original entry on oeis.org

1, 9, 99, 990, 9990, 99891, 999900, 9998901, 99998910, 999989010, 9999989010, 99999889110, 999999890109, 9999998890200, 99999998891100, 999999988901199, 9999999988902090, 99999999888912090, 999999999889011990, 9999999998889021990

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=10 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].
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:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Alois P. Heinz, Table of n, a(n) for n = 0..500

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*10^(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, Jan 25 2013

b[n_] := Sum[EulerPhi[d]*10^(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)/(1-10*x^n)  );
v=Vec(gf)

A221580 A sum over partitions (q=12), see first comment.

Original entry on oeis.org

1, 11, 143, 1716, 20724, 248677, 2985840, 35829937, 429979836, 5159757900, 61917341772, 743008099548, 8916100178843, 106993202123808, 1283918461295184, 15407021535521759, 184884258855973380, 2218611106271412996, 26623333280416468596, 319479999364994391924

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=12 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].
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:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

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

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*12^(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, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*12^(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)/(1-12*x^n)  );
v=Vec(gf)

A221581 A sum over partitions (q=14), see first comment.

Original entry on oeis.org

1, 13, 195, 2730, 38402, 537615, 7529340, 105410565, 1475786130, 20661005638, 289254613830, 4049564590890, 56693911799265, 793714765148760, 11112006817455180, 155568095444334495, 2177953337695895942, 30491346727741970070, 426878854209048054450

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=14 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].
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:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

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

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*14^(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, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*14^(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)/(1-14*x^n)  );
v=Vec(gf)

A221582 A sum over partitions (q=15), see first comment.

Original entry on oeis.org

1, 14, 224, 3360, 50610, 759136, 11390400, 170855776, 2562887040, 38443305390, 576650336640, 8649755046240, 129746337080864, 1946195056159200, 29192926013193600, 437893890197853824, 6568408355529888210, 98526125332947516960, 1477891880032655307360

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=15 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].
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:
q=6: A221578, q=10: A221579, q=12: A221580,
q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

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

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*15^(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, Feb 03 2013

b[n_] := Sum[EulerPhi[d]*15^(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)/(1-15*x^n)  );
v=Vec(gf)

A221584 A sum over partitions (q=20), see first comment.

Original entry on oeis.org

1, 19, 399, 7980, 159980, 3199581, 63999600, 1279991601, 25599991620, 511999832020, 10239999832020, 204799996632420, 4095999996640419, 81919999932640800, 1638399999932648400, 32767999998652808799, 655359999998652816380, 13107199999973052976380

Offset: 0

Joerg Arndt, Jan 20 2013

nonn

Set q=20 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].
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:
  q=6: A221578, q=10: A221579, q=12: A221580,
  q=14: A221581, q=15: A221582, q=18: A221583, q=20: A221584.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

N=66; x='x+O('x^N);
gf=prod(n=1,N, (1-x^n)/(1-20*x^n)  );
v=Vec(gf)

A319753 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)/(1 - k*x^j).

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, 1, 0, 0, 1, 2, 3, 0, 0, 1, 3, 8, 6, 0, 1, 1, 4, 15, 24, 14, 0, 0, 1, 5, 24, 60, 78, 27, 0, 1, 1, 6, 35, 120, 252, 232, 60, 0, 0, 1, 7, 48, 210, 620, 1005, 720, 117, 0, 0, 1, 8, 63, 336, 1290, 3096, 4080, 2152, 246, 0, 0, 1, 9, 80, 504, 2394, 7735, 15600, 16305, 6528, 490, 0, 0

Offset: 0

Ilya Gutkovskiy, Sep 27 2018

			Square array begins:
   1,  1,   1,    1,     1,     1,  ...
  -1,  0,   1,    2,     3,     4,  ...
  -1,  0,   3,    8,    15,    24,  ...
   0,  0,   6,   24,    60,   120,  ...
   0,  0,  14,   78,   252,   620,  ...
   1,  0,  27,  232,  1005,  3096,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..20 give A010815, A000007, A006951, A006952, A049314, A049315, A221578, A049316, A182603, A182604, A221579, A182605, A221580, A182606, A221581, A221582, A182607, A182608, A221583, A182609, A221584.

Main diagonal gives A319754.

Table[Function[k, SeriesCoefficient[Product[(1 - x^i)/(1 - k x^i), {i, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[Exp[Sum[Sum[d (k^(i/d) - 1), {d, Divisors[i]}] x^i/i, {i, n}]], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

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

G.f. of column k: exp(Sum_{j>=1} ( Sum_{d|j} d*(k^(j/d) - 1) ) * x^j/j).

cp's OEIS Frontend

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

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A221578 A sum over partitions (q=6), see first comment.

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A221579 A sum over partitions (q=10), see first comment.

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A221580 A sum over partitions (q=12), see first comment.

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A221581 A sum over partitions (q=14), see first comment.

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A221582 A sum over partitions (q=15), see first comment.

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A221584 A sum over partitions (q=20), see first comment.

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A319753 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)/(1 - k*x^j).

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