A sharp inequality for the variance with respect to the Ewens Sampling Formula

Abstract

The variance of a linear statistic defined on the symmetric group endowed with the Ewens probability is examined. Despite the dependence of the summands, it can be bounded from above by a constant multiple of the sum of variances. We find the exact value of this constant. The analysis of the appearing quadratic forms and eigenvalue search is built upon the exponential matrices and discrete Hahn's polynomials.