A000702 -id:A000702 - cp's OEIS frontend

A046682 Number of cycle types of conjugacy classes of all even permutations of n elements.

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 29, 40, 52, 69, 90, 118, 151, 195, 248, 317, 400, 505, 632, 793, 985, 1224, 1512, 1867, 2291, 2811, 3431, 4186, 5084, 6168, 7456, 9005, 10836, 13026, 15613, 18692, 22316, 26613, 31659, 37619, 44601, 52815, 62416, 73680, 86809, 102162

Offset: 0

Vladeta Jovovic

Also number of partitions of n with even number of even parts. There is no restriction on the odd parts.
a(n) = u(n) + v(n), n >= 2, of the Osima reference, p. 383.
Also number of partitions of n with largest part congruent to n modulo 2: a(2*n) = A027187(2*n), a(2*n-1) = A027193(2*n-1); a(n) = A000041(n) - A000701(n). - Reinhard Zumkeller, Apr 22 2006
Equivalently, number of partitions of n with number of parts having the same parity as n. - Olivier Gérard, Apr 04 2012
Also number of distinct free Young diagrams (Ferrers graphs with n nodes). Free Young diagrams are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another. - Jani Melik, May 08 2016
Let the cycle type of an even permutation be represented by the partition A=(O1,O2,...,Oi,E1,E2,...,E2j), where the Os are parts with odd length and the Es are parts with even lengths, and where j may be zero, using Reinhard Zumkeller's observation that the partition associated with a cycle type of an even permutation has an even number of even parts. The set of even cycle types enumerated here can be considered a monoid under a binary operation *: Let A be as above and B=(o1,o2,...,ok,e1,e2,...,e2m). A*B is the partition (O1o1,O1o2,...,O1ok,O1e1,...,O1e2m,O2o1,...,O2e2m,...,Oio1,...,Oie2m,E1o1,...,E1e2m,...,E2je2m). This product has 2im+2jk+4jm even parts, so it represents the cycle type of an even permutation. - Richard Locke Peterson, Aug 20 2018
From Gus Wiseman, Mar 31 2022: (Start)
Also the number of integer partitions of n with Heinz number greater than or equal to that of their conjugate, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are ranked by A352488. The complement is counted by A000701. For example, the a(n) partitions for n = 1...7 are:
  (1)  (11)  (21)   (22)    (221)    (222)     (331)
             (111)  (211)   (311)    (321)     (2221)
                    (1111)  (2111)   (2211)    (3211)
                            (11111)  (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)
Also the number of integer partitions of n with Heinz number less than or equal to their conjugate, ranked by A352489. For example, the a(n) partitions for n = 1...7 are:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)
            (21)  (22)  (32)   (33)   (43)
                  (31)  (41)   (42)   (52)
                        (311)  (51)   (61)
                               (321)  (322)
                               (411)  (421)
                                      (511)
                                      (4111)
(End)

			1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 12*x^8 + 16*x^9 + ...
a(3)=2 since cycle types of even permutations of 3 elements is (.)(.)(.), (...).
a(4)=3 since cycle types of even permutations of 4 elements is (.)(.)(.)(.), (...)(.), (..)(..).
a(5)=4 (free Young diagrams):
  XXXXX XXXX. XXX.. XXX..
  ..... X.... XX... X....
  ..... ..... ..... X....
  ..... ..... ..... .....
  ..... ..... ..... .....

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
J. Huh and B. Kim, The number of equivalence classes arising from partition involutions, Int. J. Number Theory, 16 (2020), 925-939.
M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.
Sheila Sundaram, On a positivity conjecture in the character table of S_n, arXiv:1808.01416 [math.CO], 2018.

Cf. A000701, A006950, A015128.
For the number of conjugacy classes of the alternating group A_n, n>=2, see A000702.
Cf. A118301.
A000041 counts integer partitions.
A000700 counts self-conjugate partitions, ranked by A088902.
A330644 counts non-self-conjugate partitions, ranked by A352486.
Heinz number (rank) and partition:
- A122111 = rank of conjugate.
- A296150 = parts of partition, conjugate A321649.
- A352487 = rank less than conjugate, counted by A000701.
- A352488 = rank greater than or equal to conjugate, counted by A046682.
- A352489 = rank less than or equal to conjugate, counted by A046682.
- A352490 = rank greater than conjugate, counted by A000701.
- A352491 = rank minus conjugate.
Cf. A114088, A115994, A171966, A238352, A258116, A321648, A325039.

seq(add((-1)^(n-k)*combinat:-numbpart(n,k),k=0..n),n=0..48); # Peter Luschny, Aug 03 2015

max = 48; f[q_] := Sum[(-q^2)^n^2, {n, 0, max}]/Product[1-q^n, {n, 1, max}]; CoefficientList[ Series[f[q], {q, 0, max}], q] (* Jean-François Alcover, Oct 18 2011, after g.f. *)
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Times@@Prime/@#>=Times@@Prime/@conj[#]&]],{n,0,15}] (* Gus Wiseman, Mar 31 2022 *)

list(lim)=my(q='q);Vec(sum(n=0,sqrt(lim),(-q^2)^(n^2))/prod(n=1,lim,1-q^n)+O(q^(lim\1+1))) \\ Charles R Greathouse IV, Oct 18 2011

{a(n) = if( n<0, 0, (numbpart(n) + polcoeff( 1 / prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n)), n)) / 2)} /* Michael Somos, Jul 24 2012 */

G.f.: Sum_{n>=0} (-q^2)^(n^2) / Product_{m>=1} (1-q^m ) = ( 1/Product_{m>=1} (1-q^m) + Product_{m>=1} (1+q^(2*m-1) ) ) / 2. - Mamuka Jibladze, Sep 07 2003
a(n) = (A000041(n) + A000700(n)) / 2.
a(n) = A000041(n) - A000701(n). - Gus Wiseman, Mar 31 2022

A124678 Number of conjugacy classes in PSL_2(p), p = prime(n).

Original entry on oeis.org

3, 4, 5, 6, 8, 9, 11, 12, 14, 17, 18, 21, 23, 24, 26, 29, 32, 33, 36, 38, 39, 42, 44, 47, 51, 53, 54, 56, 57, 59, 66, 68, 71, 72, 77, 78, 81, 84, 86, 89, 92, 93, 98, 99, 101, 102, 108, 114, 116, 117, 119, 122, 123, 128, 131, 134, 137, 138, 141, 143, 144, 149, 156, 158

Offset: 1

N. J. A. Sloane, Dec 25 2006

nonn

A great deal is known about the number of conjugacy classes in the classical linear groups. See for example Dornhoff, Section 38, or Green.

Dornhoff, Larry, Group representation theory. Part A: Ordinary representation theory. Marcel Dekker, Inc., New York, 1971.

Robin Visser, Table of n, a(n) for n = 1..10000 (terms n = 1..270 from Klaus Brockhaus).
J. A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc., 80 (1955), 402-447.

Cf. A000040, A000702, A006951, A124679, A124681.

[ NumberOfClasses(PSL(2,p)) : p in [2,3,5,7,11,13,17,19,23,29,31,37] ];

a(n) = (prime(n) + 5)/2 for all n > 1. - Robin Visser, Sep 24 2023

More terms from Klaus Brockhaus, Dec 26 2006

A124679 a(n) = number of conjugacy classes in PSL_3(prime(n)).

Original entry on oeis.org

6, 12, 30, 22, 132, 64, 306, 130, 552, 870, 334, 472, 1722, 634, 2256, 2862, 3540, 1264, 1522, 5112, 1804, 2110, 6972, 8010, 3172, 10302, 3574, 11556, 4000, 12882, 5422, 17292, 18906, 6490, 22350, 7654, 8272, 8914, 28056, 30102, 32220, 10984

Offset: 1

N. J. A. Sloane, Dec 25 2006

Jason Kimberley, Table of n, a(n) for n = 1..54
Jason Kimberley, Incomplete table of n, a(n) for n = 1..67

Cf. A000702, A006951, A124678.

A124679 := func< n | NumberOfClasses(PSL(3,NthPrime(n))) >;

a(7) to a(14) from Klaus Brockhaus, Dec 26 2006
a(15)..a(54) appended, from running MAGMA for 7 processor days at U. Newcastle, by Jason Kimberley, Feb 25 2011.
a(65)=32764 added to a124679.txt by Jason Kimberley, Mar 28 2011.

A206820 a(n) is the sum of the squares of the sizes of the conjugacy classes in the symmetric group A_n.

Original entry on oeis.org

1, 1, 3, 42, 914, 23694, 1048542, 45379878, 3272115926, 257662344206, 27726935045366, 3101635433302996, 474878584235678020, 76786899439922296204, 15844064187141655171020, 3326909755872288926885670, 897661138669999282018222470, 246381314116108359863665821750

Offset: 1

Olivier Gérard, Feb 12 2012

a(n) is the sum over all elements of Alt_n of the size of their conjugacy class. Each conjugacy class is thus counted as many times as its size, giving a sum of squares.

			For n=5, a(5) = 1 + 12^2 + 12^2 + 15^2 + 20^2 = 914.
The class equation of A_5 is  1 + 12 + 12 + 15 + 20  = 60 = 5!/2

A087132 (sequence for S_n), A000702 (conjugacy classes in A_n)

A206820 := n -> Sum(ConjugacyClasses(AlternatingGroup(n)), c->Size(c)^2); # Eric M. Schmidt, Jan 26 2014

More terms from Eric M. Schmidt, Jan 26 2014

A070733 Size of largest conjugacy class in A_n, the alternating group on n symbols.

Original entry on oeis.org

1, 1, 1, 4, 20, 90, 630, 3360, 30240, 226800, 2494800, 23950080, 311351040, 3632428800, 54486432000, 747242496000, 12703122432000, 200074178304000, 3801409387776000, 67580611338240000, 1419192838103040000, 28100018194440192000, 646300418472124416000

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 14 2002

nonn

For n > 5, the largest conjugacy class in A_n corresponds to the cycle type (n-2, 2) if n is even, (n-3, 2, 1) if n is odd. - Eric M. Schmidt, Sep 13 2014

Eric M. Schmidt, Table of n, a(n) for n = 1..100

Cf. A059171, A001710, A000702, A029726.

a:=function(n)
local G,CC,SCC,SCC1;
G:=AlternatingGroup(n);
CC:=ConjugacyClasses(G);;
SCC:=List(CC,Size);
return Maximum(SCC);
end;;  #  W. Edwin Clark, Feb 02 2014

a[n_] := (n!/2) / If[OddQ[n],  n-3, n-2]; a[1] = a[2] = a[3] = 1; a[4] = 4; a[5] = 20; Array[a, 20] (* Amiram Eldar, Jul 12 2025 *)

a(n) = if(n < 6, [1, 1, 1, 4, 20][n], (n!/2) / if(n % 2,  n-3, n-2)); \\ Amiram Eldar, Jul 12 2025

For n > 5, a(n) = n!/(2(n-2)) if n is even, a(n) = n!/(2(n-3)) if n is odd. - Eric M. Schmidt, Sep 13 2014

Sum_{n>=1} 1/a(n) = 111/10 + 1/e - 3*e. - Amiram Eldar, Jul 12 2025

More terms from Eric M. Schmidt, Sep 13 2014

A073584 Commuting even permutations: number of ordered pairs g, h in the alternating group A_n such that gh = hg.

Original entry on oeis.org

1, 1, 9, 48, 300, 2520, 22680, 282240, 3265920, 43545600, 618710400, 10298534400, 171243072000, 3138418483200, 61460695296000, 1286751578112000, 27743619391488000, 640237370572800000, 15448927751921664000, 394130125324615680000, 10422552203028725760000, 288306186674956369920000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 13 2003

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..37

Cf. A000702, A053529.

For n>=2 : a(n) = (n!/2) * A000702(n).

More terms from N. J. A. Sloane (based on A000702), Dec 31 2006

A124681 a(n) = number of conjugacy classes in PSL_4(prime(n)).

Original entry on oeis.org

14, 29, 49, 217, 757, 613, 1327, 3661, 6409, 6349, 15457, 13057, 17707, 40789, 53137, 37993, 104581, 57757, 152797

Offset: 1

N. J. A. Sloane, Dec 25 2006

Cf. A000702, A006951, A124679, A124678.

A124681 := func< n | NumberOfClasses(PSL(4,NthPrime(n))) >;

a(5) and a(6) from Klaus Brockhaus, Dec 26 2006 and  Oct 09 2010
a(7)..a(14) appended, from running MAGMA for 32 processor hours at U. Newcastle, by Jason Kimberley, Feb 09 2011.
a(15)-a(19) from Robin Visser, Oct 01 2023

A327150 Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation.

Original entry on oeis.org

1, 1, 1, 9, 22, 77, 400, 2624, 20747, 183544, 1826374, 20045348, 240262047, 3120641718, 43665293393, 654731266933, 10472819759734, 178001257647196, 3203520381407270, 60859480965537820, 1217072840308660049

Offset: 0

Derek Lim, Aug 23 2019

nonn

			For n = 3, representatives of the n=9 orbits are (e,e), (e,(123)), (e,(132)), ((123),e), ((132),e), ((123),(123)), ((123),(132)), ((132),(123)), ((132),(132)), where e is the identity.

Derek Lim, Table of n, a(n) for n = 0..61
MathOverflow, A general formula for the number of conjugacy classes of S_n×S_n acted on by S_n

Cf. A000702, A110143, A327014, A327151.

G:= AlternatingGroup(n);; Size(G)*Sum(List(ConjugacyClasses(G), K -> 1/Size(K)));

a(n) = (n!/2) * Sum_{K conjugacy class in A_n} 1/|K|.

A237036 Size of the smallest conjugacy class of size greater than 1 of the alternating group of degree n.

Original entry on oeis.org

3, 12, 40, 70, 105, 168, 240, 330, 440, 572, 728, 910, 1120, 1360, 1632, 1938, 2280, 2660, 3080, 3542, 4048, 4600, 5200, 5850, 6552, 7308, 8120, 8990, 9920, 10912, 11968, 13090, 14280, 15540, 16872, 18278, 19760, 21320, 22960, 24682, 26488, 28380, 30360, 32430

Offset: 4

W. Edwin Clark, Feb 02 2014

nonn

			For n = 4 the conjugacy classes of size greater than 1 of Alt(n) are
{(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)},
{(2,4,3), (1,2,3), (1,3,4), (1,4,2)},
{(2,3,4), (1,2,4), (1,3,2), (1,4,3)},
the smallest of which has 3 elements, hence a(4) = 3.

Wikipedia, Alternating group

Cf. A000702, A070733, A007290.

a:=function(n)
local G,CC,SCC,SCC1;
G:=AlternatingGroup(n);
CC:=ConjugacyClasses(G);;
SCC:=List(CC,Size);
SCC1:=Difference(SCC,[1]);
return Minimum(SCC1);
end;;

Join[{3,12,40,70,105},2*Binomial[Range[9,50],3]] (* Harvey P. Dale, Apr 07 2018 *)

From Alois P. Heinz, Feb 04 2014: (Start)
G.f.: -x^4*(7*x^8-28*x^7+42*x^6-20*x^5-20*x^4+30*x^3-10*x^2-3)/(x-1)^4.
a(n) = 2*C(n,3) = A007290(n) for n>=9. (End)

A371059 Number of conjugacy classes of pairs of commuting elements in the alternating group A_n.

Original entry on oeis.org

1, 1, 9, 14, 22, 44, 74, 160, 256, 462, 817, 1494, 2543, 4427, 7699, 13352, 22616, 38610, 65052, 110004, 182961, 305007, 503299, 830648, 1356227, 2212790, 3583419, 5790836

Offset: 1

Sébastien Palcoux, Mar 11 2024

The number of conjugacy classes of pairs of commuting elements in a finite group G is the cardinality of the set {c(a,b) | a,b in G and ab=ba} where c(a,b) = {(gag^(-1),gbg^(-1)) | g in G}.
It is equal to the number of conjugacy classes within the centralizers of class representatives of G.
This reformulation was employed in the sequence-generating program.
It is also equal to the rank of the modular fusion category Z(Rep(G)), the Drinfeld center of Rep(G).
These reformulations are explained in the linked MathOverflow posts.

A. Davydov, Bogomolov multiplier, double class-preserving automorphisms, and modular invariants for orbifolds. J. Math. Phys. 55 (2014), no. 9, 092305, 13 pp.

Sébastien Palcoux, Number of conjugacy classes of pairs of commuting elements, MathOverflow.
Sébastien Palcoux, Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?, MathOverflow.

Cf. A000702.

List([1..10],n->Sum(List(ConjugacyClasses(AlternatingGroup(n)),c->NrConjugacyClasses(Centralizer(AlternatingGroup(n),Representative(c))))));

cp's OEIS Frontend

A046682 Number of cycle types of conjugacy classes of all even permutations of n elements.

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A124678 Number of conjugacy classes in PSL_2(p), p = prime(n).

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A124679 a(n) = number of conjugacy classes in PSL_3(prime(n)).

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A206820 a(n) is the sum of the squares of the sizes of the conjugacy classes in the symmetric group A_n.

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A070733 Size of largest conjugacy class in A_n, the alternating group on n symbols.

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A073584 Commuting even permutations: number of ordered pairs g, h in the alternating group A_n such that gh = hg.

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A124681 a(n) = number of conjugacy classes in PSL_4(prime(n)).

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Magma

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A327150 Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation.

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