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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2511.18351 |
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| _version_ | 1866910062312685568 |
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| author | Aubert, Voalaza Mahavily Romuald Randrianirina, Benjamin |
| author_facet | Aubert, Voalaza Mahavily Romuald Randrianirina, Benjamin |
| contents | Using the lattice paths in $\mathbb{N}\times\mathbb{N}$, we derive a general formula for sequences $\big(T(n,k)\big)$ satisfying the recurrence relation of the form:
\begin{equation*}
T((n,k)=a_{n,k}T(n-1,k)+b_{n,k}T(n-1,k-1).
\end{equation*}
We apply this result to the case where $a_{n,k}=a_0+a_1k+a_2n$ and $b_{n,k}=b_0+b_1k+b_2n$.
This leads to explicit expressions for $T(n,k)$, with simpler formulas arising in the case $b_2=0$, as well as in the fully general case, using Faà di Bruno's type expression. In particular, we analyze the case $b_{n,k}=1$, which frequently occurs in enumerative combinatorics. Applications include explicit formulas for the $r$-Eulerian numbers.We also express the case $b_{n,k}=1$, using a transition matrix. We apply our results to several sequences.
\textbf{Keywords:} triangular recurrence, weighted paths, $r$-Eulerian numbers, combinatorial interpretation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18351 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit Formulas and Combinatorial Interpretation of Triangular Arrays Aubert, Voalaza Mahavily Romuald Randrianirina, Benjamin Combinatorics Using the lattice paths in $\mathbb{N}\times\mathbb{N}$, we derive a general formula for sequences $\big(T(n,k)\big)$ satisfying the recurrence relation of the form: \begin{equation*} T((n,k)=a_{n,k}T(n-1,k)+b_{n,k}T(n-1,k-1). \end{equation*} We apply this result to the case where $a_{n,k}=a_0+a_1k+a_2n$ and $b_{n,k}=b_0+b_1k+b_2n$. This leads to explicit expressions for $T(n,k)$, with simpler formulas arising in the case $b_2=0$, as well as in the fully general case, using Faà di Bruno's type expression. In particular, we analyze the case $b_{n,k}=1$, which frequently occurs in enumerative combinatorics. Applications include explicit formulas for the $r$-Eulerian numbers.We also express the case $b_{n,k}=1$, using a transition matrix. We apply our results to several sequences. \textbf{Keywords:} triangular recurrence, weighted paths, $r$-Eulerian numbers, combinatorial interpretation. |
| title | Explicit Formulas and Combinatorial Interpretation of Triangular Arrays |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.18351 |