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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2026
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2601.20770 |
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| _version_ | 1866911404573851648 |
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| author | Harris, Pamela E. Kara, Selvi McNicholas, Erin Nyman, Kathryn Yin, Mei |
| author_facet | Harris, Pamela E. Kara, Selvi McNicholas, Erin Nyman, Kathryn Yin, Mei |
| contents | We recall that a parking function of length $n+1$ is said to be prime if removing any instance of 1 yields a parking function of length $n$. In this article, we study prime parking functions from multiple lenses. We derive an explicit formula for the average value of the total displacement of prime parking functions. We present a formula for the displacement-enumerator of prime parking functions that involves a sum over Łukasiewicz paths. We describe the one-to-one correspondence between parking functions and labeledŁukasiewicz paths via Dyck paths. We introduce the concept of $\ell$-forward differences and use this as a vehicle for examining ties, ascents, and descents in prime parking functions. We establish a link between Schur functions corresponding to the partition $(i,1^{n-i})$ and fundamental quasisymmetric functions indexed by prime parking function tie sets of size $n-i.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20770 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On statistics of prime parking functions, Łukasiewicz paths, and quasisymmetric functions Harris, Pamela E. Kara, Selvi McNicholas, Erin Nyman, Kathryn Yin, Mei Combinatorics 05A05, 05A15, 05A19, 60C05 We recall that a parking function of length $n+1$ is said to be prime if removing any instance of 1 yields a parking function of length $n$. In this article, we study prime parking functions from multiple lenses. We derive an explicit formula for the average value of the total displacement of prime parking functions. We present a formula for the displacement-enumerator of prime parking functions that involves a sum over Łukasiewicz paths. We describe the one-to-one correspondence between parking functions and labeledŁukasiewicz paths via Dyck paths. We introduce the concept of $\ell$-forward differences and use this as a vehicle for examining ties, ascents, and descents in prime parking functions. We establish a link between Schur functions corresponding to the partition $(i,1^{n-i})$ and fundamental quasisymmetric functions indexed by prime parking function tie sets of size $n-i.$ |
| title | On statistics of prime parking functions, Łukasiewicz paths, and quasisymmetric functions |
| topic | Combinatorics 05A05, 05A15, 05A19, 60C05 |
| url | https://arxiv.org/abs/2601.20770 |