Enumerating several statistics of r-Colored Dyck paths with no dd-steps having the same colors
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| Format: | Preprint |
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2025
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| _version_ | 1866915335267942400 |
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| author | Sun, Yidong Wang, Jinyi Wang, Xinyu |
| author_facet | Sun, Yidong Wang, Jinyi Wang, Xinyu |
| contents | An $r$-colored Dyck path is a Dyck path with all $\mathbf{d}$-steps having one of $r$ colors in $[r]=\{1, 2, \dots, r\}$. In this paper, we consider several statistics on the set $\mathcal{A}_{n,0}^{(r)}$ of $r$-colored Dyck paths of length $2n$ with no two consecutive $\mathbf{d}$-steps having the same colors. Precisely, the paper studies the statistics ``number of points" at level $\ell$, ``number of $\mathbf{u}$-steps" at level $\ell+1$, ``number of peaks" at level $\ell+1$ and ``number of $\mathbf{udu}$-steps" on the set $\mathcal{A}_{n,0}^{(r)}$. The counting formulas of the first three statistics are established by Riordan arrays related to $S(a,b; x)$, the weighted generating function of $(a,b)$-Schröder paths. By a useful and surprising relations satisfied by $S(a,b; x)$, several identities related to these counting formulas are also described. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_08407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enumerating several statistics of r-Colored Dyck paths with no dd-steps having the same colors Sun, Yidong Wang, Jinyi Wang, Xinyu Combinatorics An $r$-colored Dyck path is a Dyck path with all $\mathbf{d}$-steps having one of $r$ colors in $[r]=\{1, 2, \dots, r\}$. In this paper, we consider several statistics on the set $\mathcal{A}_{n,0}^{(r)}$ of $r$-colored Dyck paths of length $2n$ with no two consecutive $\mathbf{d}$-steps having the same colors. Precisely, the paper studies the statistics ``number of points" at level $\ell$, ``number of $\mathbf{u}$-steps" at level $\ell+1$, ``number of peaks" at level $\ell+1$ and ``number of $\mathbf{udu}$-steps" on the set $\mathcal{A}_{n,0}^{(r)}$. The counting formulas of the first three statistics are established by Riordan arrays related to $S(a,b; x)$, the weighted generating function of $(a,b)$-Schröder paths. By a useful and surprising relations satisfied by $S(a,b; x)$, several identities related to these counting formulas are also described. |
| title | Enumerating several statistics of r-Colored Dyck paths with no dd-steps having the same colors |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.08407 |