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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

An explicit form for Kerov's character polynomials


Authors: I. P. Goulden and A. Rattan
Journal: Trans. Amer. Math. Soc. 359 (2007), 3669-3685
MSC (2000): Primary 05E10; Secondary 05A15, 20C30
Published electronically: February 23, 2007
MathSciNet review: 2302511
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Abstract: Kerov considered the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as a polynomial in free cumulants. Biane has proved that this polynomial has integer coefficients, and made various conjectures. Recently, Sniady has proved Biane's conjectured explicit form for the first family of nontrivial terms in this polynomial. In this paper, we give an explicit expression for all terms in Kerov's character polynomials. Our method is through Lagrange inversion.


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Additional Information

I. P. Goulden
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: ipgoulden@math.uwaterloo.ca

A. Rattan
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Address at time of publication: Department of Applied Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: arattan@math.mit.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04311-5
PII: S 0002-9947(07)04311-5
Received by editor(s): April 20, 2005
Published electronically: February 23, 2007
Article copyright: © Copyright 2007 American Mathematical Society