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For a distribution $p:=\{p_k\}_{k=1}^\infty$ on the positive integers, thereare 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 geometricdistribution with parameter $1-q\in(0,1)$. In this case, the $p$-shifted randompermutation 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 andthe shifted distributions on $S_n$. The expected number of inversions of apermutation 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 tohave 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 thebiased 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 ofconvergence of $q_n$ to 1. Then we consider $p$-biased and $p$-shiftedpermutations in the case that the distribution $p$ is itself random anddistributed as a GEM$(\theta)$-distribution. In both the GEM$(\theta)$-biasedand the GEM$(\theta)$-shifted cases, the expected number of inversions behavesasymptotically 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 asthe random counterpart of the Geo$(q)$-shifted case. We also consider another$p$-biased distribution with random $p$ for which the expected number ofinversions behaves asymptotically as it does under the Geo$(1-q)$-biased casewith $\theta$ and $q$ as above, and with $\theta\to\infty$ and $q\to1$.

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