Catalan numbers, parking functions, permutahedra and noncommutative Hilbert schemes
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917278423973888 |
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| author | Lunts, Valery Špenko, Špela Bergh, Michel Van den |
| author_facet | Lunts, Valery Špenko, Špela Bergh, Michel Van den |
| contents | We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $\mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of $m$-parking functions of length $n$. This bijection restricts to a bijection between the regular $S_n$-orbits and $(m,n)$-Dyck paths, the number of which is given by the Fuss-Catalan number $A_{n}(m,1)$. Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_12046 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Catalan numbers, parking functions, permutahedra and noncommutative Hilbert schemes Lunts, Valery Špenko, Špela Bergh, Michel Van den Combinatorics Algebraic Geometry Representation Theory We find an explicit $S_n$-equivariant bijection between the integral points in a certain zonotope in $\mathbb{R}^n$, combinatorially equivalent to the permutahedron, and the set of $m$-parking functions of length $n$. This bijection restricts to a bijection between the regular $S_n$-orbits and $(m,n)$-Dyck paths, the number of which is given by the Fuss-Catalan number $A_{n}(m,1)$. Our motivation came from studying tilting bundles on noncommutative Hilbert schemes. As a side result we use these tilting bundles to construct a semi-orthogonal decomposition of the derived category of noncommutative Hilbert schemes. |
| title | Catalan numbers, parking functions, permutahedra and noncommutative Hilbert schemes |
| topic | Combinatorics Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2207.12046 |