$α$-chromatic symmetric functions
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917989138300928 |
|---|---|
| author | Haglund, Jim Oh, Jaeseong Yoo, Meesue |
| author_facet | Haglund, Jim Oh, Jaeseong Yoo, Meesue |
| contents | In this paper, we introduce the \emph{$α$-chromatic symmetric functions} $χ^{(α)}_π[X;q]$, extending Shareshian and Wachs' chromatic symmetric functions with an additional real parameter $α$. We present positive combinatorial formulas with explicit interpretations. Notably, we show an explicit monomial expansion in terms of the $α$-binomial basis and an expansion into certain chromatic symmetric functions in terms of the $α$-falling factorial basis. Among various connections with other subjects, we highlight a significant link to $q$-rook theory, including a new solution to the $q$-hit problem posed by Garsia and Remmel in their 1986 paper introducing $q$-rook polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06965 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $α$-chromatic symmetric functions Haglund, Jim Oh, Jaeseong Yoo, Meesue Combinatorics 05E05, 05A19, 05A30 In this paper, we introduce the \emph{$α$-chromatic symmetric functions} $χ^{(α)}_π[X;q]$, extending Shareshian and Wachs' chromatic symmetric functions with an additional real parameter $α$. We present positive combinatorial formulas with explicit interpretations. Notably, we show an explicit monomial expansion in terms of the $α$-binomial basis and an expansion into certain chromatic symmetric functions in terms of the $α$-falling factorial basis. Among various connections with other subjects, we highlight a significant link to $q$-rook theory, including a new solution to the $q$-hit problem posed by Garsia and Remmel in their 1986 paper introducing $q$-rook polynomials. |
| title | $α$-chromatic symmetric functions |
| topic | Combinatorics 05E05, 05A19, 05A30 |
| url | https://arxiv.org/abs/2407.06965 |