Asymptotics of characters of symmetric groups related to Stanley character formula

Abstract

We prove an upper bound for characters of the symmetric groups. In particular, we show that there exists a constant a > 0 with a property that for every Young diagram λ with n boxes, r(λ) rows and c(λ) columns where is the minimal number of factors needed to write π ∈ Sn as a product of transpositions. We also give uniform estimates for the error term in the Vershik-Kerov's and Biane's character formulas and give a new formula for free cumulants of the transition measure.