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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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An explicit form for Kerov’s character polynomials
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by I. P. Goulden and A. Rattan PDF
Trans. Amer. Math. Soc. 359 (2007), 3669-3685 Request permission

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, Śniady 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
  • MR Author ID: 75735
  • 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
  • Received by editor(s): April 20, 2005
  • Published electronically: February 23, 2007
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 3669-3685
  • MSC (2000): Primary 05E10; Secondary 05A15, 20C30
  • DOI: https://doi.org/10.1090/S0002-9947-07-04311-5
  • MathSciNet review: 2302511