Set-Valued Catalan Combinatorics
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914966549823488 |
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| author | Lazar, Alexander Linusson, Svante |
| author_facet | Lazar, Alexander Linusson, Svante |
| contents | Set-valued standard Young tableaux are a generalization of standard Young tableaux due to Buch (2002) with applications in algebraic geometry. The enumeration of set-valued SYT is significantly more complicated than in the ordinary case, although product formulas are known in certain special cases. In this work we study the case of two-rowed set-valued SYT with a fixed number of entries. These tableaux are a new combinatorial model for the Catalan, Narayana, and Kreweras numbers, and can be shown to be in correspondence with both 321-avoiding permutations and a certain class of bicolored Motzkin paths. We also introduce a generalization of the set-valued comajor index studied by Hopkins, Lazar, and Linusson (2023), and use this statistic to find seemingly new q-analogs of the Catalan and Narayana numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04860 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Set-Valued Catalan Combinatorics Lazar, Alexander Linusson, Svante Combinatorics 05A05, 05A15, 05A17, 05A30 Set-valued standard Young tableaux are a generalization of standard Young tableaux due to Buch (2002) with applications in algebraic geometry. The enumeration of set-valued SYT is significantly more complicated than in the ordinary case, although product formulas are known in certain special cases. In this work we study the case of two-rowed set-valued SYT with a fixed number of entries. These tableaux are a new combinatorial model for the Catalan, Narayana, and Kreweras numbers, and can be shown to be in correspondence with both 321-avoiding permutations and a certain class of bicolored Motzkin paths. We also introduce a generalization of the set-valued comajor index studied by Hopkins, Lazar, and Linusson (2023), and use this statistic to find seemingly new q-analogs of the Catalan and Narayana numbers. |
| title | Set-Valued Catalan Combinatorics |
| topic | Combinatorics 05A05, 05A15, 05A17, 05A30 |
| url | https://arxiv.org/abs/2410.04860 |