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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2509.10868 |
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| _version_ | 1866911429730238464 |
|---|---|
| author | Musson, Ian M. |
| author_facet | Musson, Ian M. |
| contents | Let $f:\mathbb{Z}\longrightarrow \{ \times \cdot\}$ be a function such that $f(a) = \cdot$ for all except finitely for
many $a \in \mathbb{Z}$. We define a set $\flat f$ of non-intersecting arc (or cap) diagrams satisfying certain conditions determined by $f$. Then we give a recursive method for enumeration of $\flat f$ which recalls the Fundamental Recurrence for Catalan numbers. The motivation comes from the problem of enumeration of the
composition factors of a Kac module with maximum degree of atypicality for the Lie superalgebra $\mathfrak{g}=\mathfrak{gl}(r|r)$. In particular we prove a conjecture that the maximum number of
composition factors is a Catalan number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10868 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Catalan numbers and a conjecture on the maximum composition length of a Kac module Musson, Ian M. Combinatorics 05E10, 17B10 Let $f:\mathbb{Z}\longrightarrow \{ \times \cdot\}$ be a function such that $f(a) = \cdot$ for all except finitely for many $a \in \mathbb{Z}$. We define a set $\flat f$ of non-intersecting arc (or cap) diagrams satisfying certain conditions determined by $f$. Then we give a recursive method for enumeration of $\flat f$ which recalls the Fundamental Recurrence for Catalan numbers. The motivation comes from the problem of enumeration of the composition factors of a Kac module with maximum degree of atypicality for the Lie superalgebra $\mathfrak{g}=\mathfrak{gl}(r|r)$. In particular we prove a conjecture that the maximum number of composition factors is a Catalan number. |
| title | Catalan numbers and a conjecture on the maximum composition length of a Kac module |
| topic | Combinatorics 05E10, 17B10 |
| url | https://arxiv.org/abs/2509.10868 |