Asymptotic Normality of Statistics on Permutation Tableaux

TL;DR

This paper derives exact characteristic functions for statistics on permutation tableaux, establishing their asymptotic normality and providing precise distributional insights through a probabilistic approach.

Contribution

It introduces a probabilistic method to derive exact distributions and proves asymptotic normality for multiple statistics on permutation tableaux, including a new result for superfluous 1s.

Findings

Distributions of basic permutation tableau statistics are exactly identified.

Number of superfluous 1s is asymptotically normal.

Established a general condition for CLT using dependency graphs.

Abstract

In this paper we use a probabilistic approach to derive the expressions for the characteristic functions of basic statistics defined on permutation tableaux. Since our expressions are exact, we can identify the distributions of basic statistics (like the number of unrestricted rows, the number of rows, and the number of 1s in the first row) exactly. In all three cases the distributions are known to be asymptotically normal after a suitable normalization. We also establish the asymptotic normality of the number of superfluous 1s. The latter result relies on a bijection between permutation tableaux and permutations and on a rather general sufficient condition for the central limit theorem for the sums of random variables in terms of dependency graph of the summands.