A subfamily of skew Dyck paths related to $k$-ary trees
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909077696675840 |
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| author | Zhang, Yuxuan Zhuang, Yan |
| author_facet | Zhang, Yuxuan Zhuang, Yan |
| contents | We introduce a subfamily of skew Dyck paths called box paths and show that they are in bijection with pairs of ternary trees, confirming an observation stated previously on the On-Line Encyclopedia of Integer Sequences. More generally, we define $k$-box paths, which are in bijection with $(k+1)$-tuples of $(k+2)$-ary trees. A bijection is given between $k$-box paths and a subfamily of $k_{t}$-Dyck paths, as well as a bijection with a subfamily of $(k,\ell)$-threshold sequences. We also study the refined enumeration of $k$-box paths by the number of returns and the number of long ascents. Notably, the distribution of long ascents over $k$-box paths generalizes the Narayana distribution on Dyck paths, and we find that $(k-3)$-box paths with exactly two long ascents provide a combinatorial model for the second $k$-gonal numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15778 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A subfamily of skew Dyck paths related to $k$-ary trees Zhang, Yuxuan Zhuang, Yan Combinatorics 05A15 (Primary), 05A10, 05A19 (Secondary) We introduce a subfamily of skew Dyck paths called box paths and show that they are in bijection with pairs of ternary trees, confirming an observation stated previously on the On-Line Encyclopedia of Integer Sequences. More generally, we define $k$-box paths, which are in bijection with $(k+1)$-tuples of $(k+2)$-ary trees. A bijection is given between $k$-box paths and a subfamily of $k_{t}$-Dyck paths, as well as a bijection with a subfamily of $(k,\ell)$-threshold sequences. We also study the refined enumeration of $k$-box paths by the number of returns and the number of long ascents. Notably, the distribution of long ascents over $k$-box paths generalizes the Narayana distribution on Dyck paths, and we find that $(k-3)$-box paths with exactly two long ascents provide a combinatorial model for the second $k$-gonal numbers. |
| title | A subfamily of skew Dyck paths related to $k$-ary trees |
| topic | Combinatorics 05A15 (Primary), 05A10, 05A19 (Secondary) |
| url | https://arxiv.org/abs/2306.15778 |