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

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