

A087132


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


4



1, 2, 14, 146, 2602, 71412, 2675724, 134269158, 8747088662, 717107850956, 72007758701716, 8736187050160132, 1258160557017484564, 212232765513231245096, 41518913481377118146520, 9309797624034705006898470, 2374942651509463493006400390, 683620331016710787068868581580
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OFFSET

1,2


COMMENTS

This is a natural quantity to consider when viewing the symmetric group (Sym_n) as a set. a(n) is the sum over all elements of Sym_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.  Olivier Gérard, Feb 12 2012


LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 1..57
Simon R. Blackburn, John R. Britnell, Mark Wildon, The probability that a pair of elements of a finite group are conjugate, arXiv:1108.1784, 2011
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math.CO/0606370


FORMULA

a(n) = (n!)^2 * (c/n^2 + O((log n)/n^3)), where c = prod_{k>=1}sum_{n>=0}1/(k*n!)^2 ~ 4.263403514152669778298935... (see A246879). [Corrected by Vaclav Kotesovec, Sep 21 2014]


PROG

(MAGMA) [ &+[ c[2]^2 : c in ClassesData(Sym(n))] : n in [1..10]]; // Sergei Haller (sergei(AT)sergeihaller.de), Dec 21 2006


CROSSREFS

Cf. A000041, A073906, A192983, A206820.  Olivier Gérard, Feb 12 2012
Cf. A246879.
Sequence in context: A328004 A271564 A100510 * A036079 A121227 A250916
Adjacent sequences: A087129 A087130 A087131 * A087133 A087134 A087135


KEYWORD

nonn


AUTHOR

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 18 2003


EXTENSIONS

More terms from Vladeta Jovovic, Oct 22 2003
More terms from Vaclav Kotesovec, Sep 21 2014


STATUS

approved



