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

We introduce a rather natural family of non-uniform distributions on $P{F}_n$, $n\in \mathbb{N}$, 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 $(\sigma ,\tau )\to ({\tilde{I}}_{n,<{\tau }_1}(\sigma ),\cdots ,{\tilde{I}}_{n,<{\tau }_n}(\sigma ))$ maps $S_n\times S_n$ onto $P{F}_n$. Consider the family of distributions $P_n^{(q)}\times P_n$, $q\in (0,\mathrm{\infty })$, 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 $P{F}_n$. We study the asymptotic behavior of two fundamental statistics of parking functions under the family of distributions ${\mathcal{P}}_n^{(q)}$.