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 | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9947-07-04311-5

Received by editor(s):
April 20, 2005

Published electronically:
February 23, 2007

Article copyright:
© Copyright 2007
American Mathematical Society