Catalan numbers: from FC elements to classical diagram algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917569037860864 |
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| author | Harbat, Sadek Al |
| author_facet | Harbat, Sadek Al |
| contents | Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-type Coxeter group. Using only the settings of their canonical form, we recount $W^c(A_n)$ by the recurrence that is taken as a definition of the Catalan number $C_{n+1}$ and we find the Narayana numbers as well as the Catalan triangle via suitable set partitions of $W^c(A_n)$. We determine the unique bijection between $W^c(A_n)$ and the set of non-crossing diagrams of $n+1$ strings that respects the diagrammatic multiplication by concatenation in the $A_n$-type Temperley-Lieb algebra, along with the two algorithms implementing this bijection and its inverse. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_10100 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Catalan numbers: from FC elements to classical diagram algebras Harbat, Sadek Al Combinatorics Representation Theory 05E10, 05Axx, Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-type Coxeter group. Using only the settings of their canonical form, we recount $W^c(A_n)$ by the recurrence that is taken as a definition of the Catalan number $C_{n+1}$ and we find the Narayana numbers as well as the Catalan triangle via suitable set partitions of $W^c(A_n)$. We determine the unique bijection between $W^c(A_n)$ and the set of non-crossing diagrams of $n+1$ strings that respects the diagrammatic multiplication by concatenation in the $A_n$-type Temperley-Lieb algebra, along with the two algorithms implementing this bijection and its inverse. |
| title | Catalan numbers: from FC elements to classical diagram algebras |
| topic | Combinatorics Representation Theory 05E10, 05Axx, |
| url | https://arxiv.org/abs/2308.10100 |