Comparing the inversion statistic for distribution-biased and distribution-shifted permutations with the geometric and the GEM distributions

Abstract

Given a probability distribution $p:=\{p_k\}_{k=1}^\infty$ on the positive integers, there are two natural ways to construct a random permutation of $\mathbb{N}$. One is called the $p$-biased construction and the other the $p$-shifted construction. For any $n\in\mathbb{N}$, amending these constructions in an obvious way yields $p$-biased and $p$-shifted random permutations of $[n]$, that is distributions on the set $S_n$ of permutation of $[n]$. In the first part of the paper we consider the case that the distribution $p$ is the geometric distribution with parameter $1-q\in(0,1)$. In this case, the $p$-shifted random permutation has the Mallows distribution with parameter $q$. Let $P_n^{b;\text{Geo}(1-q)}$ and $P_n^{s;\text{Geo}(1-q)}$ denote the biased and the shifted distributions on $S_n$. The number of inversions of a permutation under $P_n^{s;\text{Geo}(1-q)}$ stochastically dominates the number of inversions under $P_n^{b;\text{Geo}(1-q)}$, and under either of these distributions, a permutation tends to have many fewer inversions than it would have under the uniform distribution. For fixed $n$, both $P_n^{b;\text{Geo}(1-q)}$ and $P_n^{s;\text{Geo}(1-q)}$ converge weakly as $q\to1$ to the uniform distribution on $S_n$. We compare the biased and the shifted distributions by studying the inversion statistic under $P_n^{b;\text{Geo}(q_n)}$ and $P_n^{s;\text{Geo}(q_n)}$ for various rates of convergence of $q_n$ to 1. In the second part of the paper we consider $p$-biased and $p$-shifted permutations for the case that the distribution $p$ is itself random and distributed as a GEM$(\theta)$-distribution. In particular, in both the GEM$(\theta)$-biased and the GEM$(\theta)$-shifted cases, the expected number of inversions behaves asymptotically like the expected number of inversions under the Geo$(1-q)$-shifted distribution with $\theta=\frac q{1-q}$.