Luck and magic for Pitman-Stanley polytopes and parking functions
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| Format: | Preprint |
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2026
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| _version_ | 1866913014915006464 |
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| author | Avila, Nicolas Ferroni, Luis Morales, Alejandro H. |
| author_facet | Avila, Nicolas Ferroni, Luis Morales, Alejandro H. |
| contents | Motivated by the combinatorics of parking functions and their several generalizations, we study the Ehrhart theory of Pitman--Stanley polytopes. We prove a strong positivity phenomenon called \emph{magic positivity} for the Ehrhart polynomials of these polytopes, which in turn implies that their $h^*$-polynomials are real-rooted (and thus log-concave and unimodal). Our result is achieved by interpreting the coefficients of these Ehrhart polynomials in the \emph{magic basis} in terms of the number of \emph{lucky cars} in a modified parking protocol. Furthermore, we address the magic positivity problem for $\mathbf{y}$-generalized permutohedra and also discuss a \emph{magic} combinatorial interpretation for them, under the assumption that the input parameters are sufficiently large. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_19194 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Luck and magic for Pitman-Stanley polytopes and parking functions Avila, Nicolas Ferroni, Luis Morales, Alejandro H. Combinatorics Primary: 05A05, 05A19, 52A38, 11B83, Secondary: 52B05, 52B11, 52B12 Motivated by the combinatorics of parking functions and their several generalizations, we study the Ehrhart theory of Pitman--Stanley polytopes. We prove a strong positivity phenomenon called \emph{magic positivity} for the Ehrhart polynomials of these polytopes, which in turn implies that their $h^*$-polynomials are real-rooted (and thus log-concave and unimodal). Our result is achieved by interpreting the coefficients of these Ehrhart polynomials in the \emph{magic basis} in terms of the number of \emph{lucky cars} in a modified parking protocol. Furthermore, we address the magic positivity problem for $\mathbf{y}$-generalized permutohedra and also discuss a \emph{magic} combinatorial interpretation for them, under the assumption that the input parameters are sufficiently large. |
| title | Luck and magic for Pitman-Stanley polytopes and parking functions |
| topic | Combinatorics Primary: 05A05, 05A19, 52A38, 11B83, Secondary: 52B05, 52B11, 52B12 |
| url | https://arxiv.org/abs/2603.19194 |