A positivity conjecture for Kerov polynomials

Kerov polynomials express the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as polynomials in the “free cumulants” of the associated Young diagram. We present a positivity conjecture for their coefficients. The latter is stronger than the positivity conjecture due to Kerov and Biane. 1 Kerov polynomials 1.1 Characters A partition λ = (λ1, ..., λr) is a finite weakly decreasing sequence of nonnegative integers, called parts. The number l(λ) of positive parts is called the length of λ, and |λ| = ∑r i=1 λi the weight of λ. For any integer i ≥ 1, mi(λ) = card{j : λj = i} is the multiplicity of the part i in λ. Let n be a fixed positive integer and Sn the group of permutations of n letters. Each permutation σ ∈ Sn factorizes uniquely as a product of disjoint cycles, whose respective lengths are ordered such as to form a partition μ = (μ1, . . . , μr) with weight n, the socalled cycle-type of σ. The irreducible representations of Sn and their corresponding characters are also labelled by partitions λ with weight |λ| = n. We write dimλ for the dimenion of the representation λ and χμ for the value of the character χ (σ) at any permutation σ of cycle-type μ. Let r ≤ n be a positive integer and μ = (r, 1) the corresponding r-cycle in Sn. We write χ̂λr = n(n− 1) · · · (n− r + 1) χ r,1 dimλ for the value at μ of the normalized character. It was first observed by Kerov[5] and Biane[2] that χ̂λr may be written as a polynomial in the “free cumulants” of the Young diagram of λ. 1 1.2 Free cumulants Two increasing sequences y = (y1, . . . , yd−1) and x = (x1, . . . , xd−1, xd) are said to be interlacing if x1 < y1 < x2 < · · · < xd−1 < yd−1 < xd. The center of the pair is c(x, y) = ∑