A family of non-uniform distributions on the set of parking functions generated by random permutations
Ross G. Pinsky
TL;DR
The paper introduces a natural, parametric family of non-uniform, exchangeable distributions on parking functions $PF_n$ by applying Mallows tilting to permutation pairs and transporting via a Lehmer/Lehmer-like code. It then analyzes two primary statistics under these measures: the first-coordinate entry $\pi_1$ of a parking function and the occupancy counts $N^{(n)}_k$ of values, deriving a rich set of asymptotics across multiple regimes for the Mallows parameter $q_n$. Key results include explicit formulas and monotonicity for the distribution of $\pi_1$, with limits ranging from geometric (for $q<1$) to uniform (for certain $q>1$ regimes), as well as Poisson and Bernoulli-type limits for $N^{(n)}_k$ under near-1, subcritical, and supercritical tilts. The work connects to analogous partition-model measures and provides insights into when the non-uniform parking-function models approximate the uniform case, with potential implications for simulations and probabilistic modeling of parking functions.
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 $σ\in S_n$ and for $j\in[n]$, let $I_{n,<j}(σ)$ denote the number of inversions in $σ$ that involve the number $j$ and a number less than $j$. Let $\tilde I_{n,<j}(σ)=I_{n,<j}(σ)+1$. The map $(σ,τ)\to\left(\tilde I_{n,<τ_1}(σ),\cdots, \tilde I_{n,< τ_n}(σ)\right)$ 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 $\mathcal{P}_n^{(q)}$.