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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.22719 |
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| _version_ | 1866911083835424768 |
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| author | Caldwell, Drew Cochran, Ali Glisson, Nathan Jennings, Bryce McDicken, Katy Proctor, Luke Klanderman, Sarah Tebbe, Amelia |
| author_facet | Caldwell, Drew Cochran, Ali Glisson, Nathan Jennings, Bryce McDicken, Katy Proctor, Luke Klanderman, Sarah Tebbe, Amelia |
| contents | Action graphs emerged from work of Bergner and Hackney on category actions in the context of Reedy categories. Alvarez, Bergner, and Lopez showed that action graphs could be inductively generated without reference to category actions and have a close relationship with the sequence of Catalan numbers. These graphs were further generalized in work of Cressman, Lin, Nguyen, and Wiljanen, who showed that the Fuss-Catalan numbers have a similar relation to another set of inductively defined directed graphs. In our paper, we consider several other sequences related to the Catalan numbers, namely Catalan's triangle, $(a,b)$-Catalan numbers, internal triangles, and super Catalan numbers. We show action graphs cannot be generalized to Catalan's triangle, $(a,b)$-Catalan numbers, nor internal triangles. We also conjecture a method for constructing action graphs for the Super Catalan numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Catalan number sequences and generalized action graphs Caldwell, Drew Cochran, Ali Glisson, Nathan Jennings, Bryce McDicken, Katy Proctor, Luke Klanderman, Sarah Tebbe, Amelia Combinatorics 05C20 (Primary) 05A19, 05C05 (Secondary) Action graphs emerged from work of Bergner and Hackney on category actions in the context of Reedy categories. Alvarez, Bergner, and Lopez showed that action graphs could be inductively generated without reference to category actions and have a close relationship with the sequence of Catalan numbers. These graphs were further generalized in work of Cressman, Lin, Nguyen, and Wiljanen, who showed that the Fuss-Catalan numbers have a similar relation to another set of inductively defined directed graphs. In our paper, we consider several other sequences related to the Catalan numbers, namely Catalan's triangle, $(a,b)$-Catalan numbers, internal triangles, and super Catalan numbers. We show action graphs cannot be generalized to Catalan's triangle, $(a,b)$-Catalan numbers, nor internal triangles. We also conjecture a method for constructing action graphs for the Super Catalan numbers. |
| title | Catalan number sequences and generalized action graphs |
| topic | Combinatorics 05C20 (Primary) 05A19, 05C05 (Secondary) |
| url | https://arxiv.org/abs/2507.22719 |