Parking functions, Fubini rankings, and Boolean intervals in the weak order of $\mathfrak{S}_n$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916801198161920 |
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| author | Elder, Jennifer Harris, Pamela E. Kretschmann, Jan Mori, J. Carlos Martínez |
| author_facet | Elder, Jennifer Harris, Pamela E. Kretschmann, Jan Mori, J. Carlos Martínez |
| contents | Let $\mathfrak{S}_n$ denote the symmetric group and let $W(\mathfrak{S}_n)$ denote the weak order of $\mathfrak{S}_n$. Through a surprising connection to a subset of parking functions, which we call unit Fubini rankings, we provide a complete characterization and enumeration for the total number of Boolean intervals in $W(\mathfrak{S}_n)$ and the total number of Boolean intervals of rank $k$ in $W(\mathfrak{S}_n)$. Furthermore, for any $π\in\mathfrak{S}_n$, we establish that the number of Boolean intervals in $W(\mathfrak{S}_n)$ with minimal element $π$ is a product of Fibonacci numbers. We conclude with some directions for further study. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_14734 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Parking functions, Fubini rankings, and Boolean intervals in the weak order of $\mathfrak{S}_n$ Elder, Jennifer Harris, Pamela E. Kretschmann, Jan Mori, J. Carlos Martínez Combinatorics 05A05, 06A07, 05A05, 05A15, 05A19 Let $\mathfrak{S}_n$ denote the symmetric group and let $W(\mathfrak{S}_n)$ denote the weak order of $\mathfrak{S}_n$. Through a surprising connection to a subset of parking functions, which we call unit Fubini rankings, we provide a complete characterization and enumeration for the total number of Boolean intervals in $W(\mathfrak{S}_n)$ and the total number of Boolean intervals of rank $k$ in $W(\mathfrak{S}_n)$. Furthermore, for any $π\in\mathfrak{S}_n$, we establish that the number of Boolean intervals in $W(\mathfrak{S}_n)$ with minimal element $π$ is a product of Fibonacci numbers. We conclude with some directions for further study. |
| title | Parking functions, Fubini rankings, and Boolean intervals in the weak order of $\mathfrak{S}_n$ |
| topic | Combinatorics 05A05, 06A07, 05A05, 05A15, 05A19 |
| url | https://arxiv.org/abs/2306.14734 |