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

For a distribution $p:=\{p_k{\}}_{k=1}^{\mathrm{\infty }}$ 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 $q\to 1$ 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 $\theta \to \mathrm{\infty }$ and $q\to 1$.