Amarpreet Rattan

An explicit form for Kerov's character polynomials

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 Bianes 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 Kerovs character polynomials. Our method is through Lagrange inversion.

Minimal factorizations of permutations into star transpositions

We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form (1i). This generalizes earlier work of Pak in which substantial restrictions were placed on the permutation being factored. Our result exhibits an unexpected and simple symmetry of star factorizations that has yet to be explained in a satisfactory manner.

Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula

We study asymptotics of an irreducible representation of the symmetric group Sn corresponding to a balanced Young diagram ? (a Young diagram with at most rows and columns for some fixed constant C) in the limit as n tends to infinity. We show that there exists a constant D (which depends only on C) with a property that where |p| denotes the length of a permutation (the minimal number of factors necessary to write p as a product of transpositions). Our main tool is an analogue of the Frobenius character formula which holds true not only for cycles but for arbitrary permutations.

Stanley's character polynomials and coloured factorisations in the symmetric group

In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, Sem. Lothar. Combin. 50 (2003) B50d, 11 p.] the author introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. In [R.P. Stanley, A conjectured combinatorial interpretation of the normalised irreducible character values of the symmetric group, math.CO/0606467, 2006] the same author gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree.