Asymptotic independence in large random permutations with fixed descent set

Abstract

Ehrenborg, Levin and Readdy have introduced a new probabilistic approachto the combinatorics of permutations with fixed set of descents. In this paper we extend this approach by introducing a more general probabilistic model. The study ofthis model yields new estimates on the behavior of a uniform random permutation σhaving a fixed descent set. In particular, we find a positive answer to a conjecture and we show that independently of the shape of the descent set, $σ(i)$ and $σ(j)$ are almost independent when $i − j$ becomes large.