A000702 a(n) is the number of conjugacy classes in the alternating group A_n.
1, 1, 3, 4, 5, 7, 9, 14, 18, 24, 31, 43, 55, 72, 94, 123, 156, 200, 254, 324, 408, 513, 641, 804, 997, 1236, 1526, 1883, 2308, 2829, 3451, 4209, 5109, 6194, 7485, 9038, 10871, 13063, 15654, 18738, 22365, 26665, 31716, 37682, 44669, 52887, 62494, 73767
Offset: 1
Examples
G.f. = x + x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 9*x^7 + 14*x^8 + 18*x^9 + ...
References
- Girse, Robert D.; The number of conjugacy classes of the alternating group. Nordisk Tidskr. Informationsbehandling (BIT) 20 (1980), no. 4, 515-517.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- T. D. Noe, Table of n, a(n) for n = 1..1000 [a(1)=1 prepended by _Georg Fischer_, Sep 29 2020]
- R. D. Girse, The number of conjugacy classes of the alternating group, Preprint and correspondence [Annotated scanned copy]
- M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.
- Eric Weisstein's World of Mathematics, Alternating Group.
- Index entries for sequences related to groups
Crossrefs
Cf. A073584.
Programs
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Magma
[ NumberOfClasses(Alt(n)) : n in [1..10] ]; /* A useful example of MAGMA code, but it is better to use one of the formulas as below: */ A000702:= func< n | 2*NumberOfPartitions(n)+3*(&+[(-1)^r*NumberOfPartitions(n-2*r^2): r in [1..Isqrt(n div 2)]]) >; [1] cat [A000702(n): n in [2..48]]; // Jason Kimberley, Feb 01 2011
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Mathematica
p = PartitionsP; q[n_] := SeriesCoefficient[ Product[ 1+x^(2k+1), {k, 0, n}], {x, 0, n}]; a[1]=1; a[n_] := (p[n] + 3*q[n])/2; Table[a[n], {n, 48}] (* Jean-François Alcover, Feb 22 2012, after first formula *) a[ n_] := SeriesCoefficient[ ( 1 / QPochhammer[ x] + 3 / QPochhammer[ x, -x] ) / 2 - (2 + x), {x, 0, n}]; Table[a[n], {n, 1, 48}] (* Michael Somos, May 28 2014 *) -
PARI
default(seriesprecision,99); Vec((1/eta(x)+3*eta(x^2)^2/(eta(x)*eta(x^4)))/2-(2+x)) /* Joerg Arndt, Feb 02 2011 */
Formula
a(n) = (p(n) + 3Q(n))/2 for n>1 where p(n) denotes the number of unrestricted partitions of n (A000041) and Q(n) the number of partitions of n into distinct odd parts (A000700). [Denes-Erdős-Turan]
a(n) = 2p(n) + 3*Sum_{r>=1} (-1)^r*p(n-2r^2) for n>1. [Girse]
Sum_{r>=0} (-1)^r*a(n-(3r^2 +- r)/2) = 3(-1)^t if n = 2t^2 or 0 otherwise, where p(u) and a(u) are taken as 0 unless u is a nonnegative integer and t = 1,2,3,... [Girse]
Extensions
a(1)=1 added by N. J. A. Sloane, Sep 14 2020
Follow-up corrections by Andrey Zabolotskiy, Sep 18 2020