On some discrete statistics of parking functions
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arXiv
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916259552034816 |
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| author | Cruz, Ari Harris, Pamela E. Harry, Kimberly J. Kretschmann, Jan McClinton, Matt Moon, Alex Museus, John O. Redmon, Eric |
| author_facet | Cruz, Ari Harris, Pamela E. Harry, Kimberly J. Kretschmann, Jan McClinton, Matt Moon, Alex Museus, John O. Redmon, Eric |
| contents | Recall that $α=(a_1,a_2,\ldots,a_n)\in[n]^n$ is a parking function if its nondecreasing rearrangement $β=(b_1,b_2,\ldots,b_n)$ satisfies $b_i\leq i$ for all $1\leq i\leq n$. In this article, we study parking functions based on their ascents (indices at which $a_i<a_{i+1}$), descents (indices at which $a_i>a_{i+1}$), and ties (indices at which $a_i=a_{i+1}$). By utilizing multiset Eulerian polynomials, we give a generating function for the number of parking functions of length $n$ with $i$ descents. We present a recursive formula for the number of parking functions of length $n$ with descents at a specified subset of $[n-1]$. We establish that the number of parking functions of length $n$ with descents at $I\subset[n-1]$ and descents at $J=\{n-i:i\in I\}$ are equinumerous. As a special case, we show that the number of parking functions of length $n$ with descents at the first $k$ indices is given by $f(n, n-k-1)=\frac{1}{n}\binom{n}{k}\binom{2n-k}{n-k-1}$. We prove this by bijecting to the set of standard Young tableaux of shape $((n-k)^2,1^k)$, which are enumerated by $f(n,n-k-1)$. We also study peaks of parking functions, which are indices at which $a_{i-1}<a_i>a_{i+1}$. We show that the set of parking functions with no peaks and no ties is enumerated by the Catalan numbers. We conclude our study by characterizing when a parking function is uniquely determined by their statistic encoding; a word indicating what indices in the parking function are ascents, descents, and ties. We provide open problems throughout. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16786 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On some discrete statistics of parking functions Cruz, Ari Harris, Pamela E. Harry, Kimberly J. Kretschmann, Jan McClinton, Matt Moon, Alex Museus, John O. Redmon, Eric Combinatorics 05A05, 05A15 Recall that $α=(a_1,a_2,\ldots,a_n)\in[n]^n$ is a parking function if its nondecreasing rearrangement $β=(b_1,b_2,\ldots,b_n)$ satisfies $b_i\leq i$ for all $1\leq i\leq n$. In this article, we study parking functions based on their ascents (indices at which $a_i<a_{i+1}$), descents (indices at which $a_i>a_{i+1}$), and ties (indices at which $a_i=a_{i+1}$). By utilizing multiset Eulerian polynomials, we give a generating function for the number of parking functions of length $n$ with $i$ descents. We present a recursive formula for the number of parking functions of length $n$ with descents at a specified subset of $[n-1]$. We establish that the number of parking functions of length $n$ with descents at $I\subset[n-1]$ and descents at $J=\{n-i:i\in I\}$ are equinumerous. As a special case, we show that the number of parking functions of length $n$ with descents at the first $k$ indices is given by $f(n, n-k-1)=\frac{1}{n}\binom{n}{k}\binom{2n-k}{n-k-1}$. We prove this by bijecting to the set of standard Young tableaux of shape $((n-k)^2,1^k)$, which are enumerated by $f(n,n-k-1)$. We also study peaks of parking functions, which are indices at which $a_{i-1}<a_i>a_{i+1}$. We show that the set of parking functions with no peaks and no ties is enumerated by the Catalan numbers. We conclude our study by characterizing when a parking function is uniquely determined by their statistic encoding; a word indicating what indices in the parking function are ascents, descents, and ties. We provide open problems throughout. |
| title | On some discrete statistics of parking functions |
| topic | Combinatorics 05A05, 05A15 |
| url | https://arxiv.org/abs/2312.16786 |