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

TL;DR

This paper compares inversion statistics of distribution-biased and distribution-shifted permutations, focusing on geometric and GEM distributions, and analyzes their asymptotic behaviors and convergence properties.

Contribution

It introduces a detailed comparison of biased and shifted permutation models for geometric and GEM distributions, highlighting their asymptotic inversion behaviors and convergence to uniform distribution.

Findings

Shifted permutations have more inversions than biased ones under geometric distribution.

Both models converge to uniform distribution as parameters approach certain limits.

GEM-distributed cases behave asymptotically like geometric cases with specific parameter relations.

Abstract

For a distribution $p:=\{p_k\}_{k=1}^\infty$ on the positive integers, there are two natural ways to construct a random permutation in $S_n$ or of $\mathbb{N}$ from IID samples from $p$--the $p$-biased construction and the $p$-shifted construction. First we consider the case that $p$ 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 expected number of inversions of a permutation under $P_n^{s;\text{Geo}(1-q)}$ is greater than under $P_n^{b;\text{Geo}(1-q)}$, and under either of these, 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. Then we consider $p$-biased and $p$-shifted permutations in the case that the distribution $p$ is itself random and distributed as a GEM$(\theta)$-distribution. In both the GEM$(\theta)$-biased and the GEM$(\theta)$-shifted cases, the expected number of inversions behaves asymptotically as it does under the Geo$(1-q)$-shifted distribution with $\theta=\frac q{1-q}$. Thus, one can consider the GEM$(\theta)$-shifted case as the random counterpart of the Geo$(q)$-shifted case. We also consider another $p$-biased distribution with random $p$ for which the expected number of inversions behaves asymptotically as it does under the Geo$(1-q)$-biased case with $\theta$ and $q$ as above, and with $\theta\to\infty$ and $q\to1$.