Major Index Distribution

Abstract

For $0<q<1$, let $Maj$ be the distribution on the symmetric group $S_n$ such that a permutation $\pi \in S_n$ is selected with probability proportional to $q^{maj(\pi)}$. The distribution has connections to $q$-Plancherel measure. We describe an algorithm that realizes $Maj$, and use it to prove known results of $q$-Plancherel measure without the need of representation theory. This sampler is transparent and elegant, allowing properties of $Maj$ about its limit shape, pattern normality, and cycle structure to be obtained.