A family of non-uniform distributions on the set of parking functions generated by random permutations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915533422592000 |
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| author | Pinsky, Ross G. |
| author_facet | Pinsky, Ross G. |
| contents | 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 $\mathcal{P}_n^{(q)}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_24452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A family of non-uniform distributions on the set of parking functions generated by random permutations Pinsky, Ross G. Probability Combinatorics 60C05, 05A05 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 $\mathcal{P}_n^{(q)}$. |
| title | A family of non-uniform distributions on the set of parking functions generated by random permutations |
| topic | Probability Combinatorics 60C05, 05A05 |
| url | https://arxiv.org/abs/2509.24452 |