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 PFₙ, 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∈ Sₙ and for j∈[n], let In,<j(σ) denote the number of inversions in $σ$ that involve the number j and a number less than j. Let In,<j(σ)=In,<j(σ)+1. The map (σ,τ)→( In,<τ₁(σ),⋯, In,< τₙ(σ)) maps Sₙ× Sₙ onto PFₙ. Consider the family of distributions Pₙ(q)× Pₙ, q∈(0,∞), on Sₙ× Sₙ, where Pₙ is the uniform distribution on Sₙ and Pₙ(q) is the Mallows distribution with parameter q on Sₙ. The Mallows distributions are defined by exponential tilting via the inversion statistic. For each $q>0$, the above map along with the distribution Pₙ(q)× Pₙ induces an exchangeable distribution Pₙ(q) on PFₙ. We study the asymptotic behavior of two fundamental statistics of parking functions under the family of distributions Pₙ(q).