A family of non-uniform distributions on the set of parking functions generated by random permutations

Abstract

We introduce a rather natural family of non-uniform distributions on $PF_n$, $n\in\mathbb{N}$, the set of parking functions of length $n$. One of the motivations for this comes from a similar situation in the context of integer partitions. For a permutation $\sigma\in S_n$ and for $j\in[n]$, let $I_{n,<j}(\sigma)$ denote the number of inversions in $\sigma$ that involve the number $j$ and a number less than $j$. Let $\tilde I_{n,<j}(\sigma)=I_{n,<j}(\sigma)+1$. The map $(\sigma,\tau)\to\left(\tilde I_{n,<\tau_1}(\sigma),\cdots, \tilde I_{n,< \tau_n}(\sigma)\right)$ maps $S_n\times S_n$ onto $PF_n$. Consider the family of distributions $P_n^{(q)}\times P_n$, $q\in(0,\infty)$, on $S_n\times S_n$, where $P_n$ is the uniform distribution on $S_n$ and $P_n^{(q)}$ is the Mallows distribution with parameter $q$ on $S_n$. The Mallows distributions are defined by exponential tilting via the inversion statistic. For each $q>0$, the above map along with the distribution $P_n^{(q)}\times P_n$ induces an exchangeable distribution $\mathcal{P}_n^{(q)}$ on $PF_n$. We study the asymptotic behavior of two fundamental statistics of parking functions under the family of distributions $\mathcal{P}_n^{(q)}$.