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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2512.01005 |
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| _version_ | 1866913070324908032 |
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| author | Aubert, Voalaza Mahavily Romuald Randrianirina, Benjamin |
| author_facet | Aubert, Voalaza Mahavily Romuald Randrianirina, Benjamin |
| contents | In this work, the Hao grammar $G=\{\, u\rightarrow u^{b_1+b_2+1} v^{a_1+a_2},\quad
v\rightarrow u^{b_2}v^{a_2+1} \,\},$
together with the correspondence between grammars and combinatorial differential equations, is employed to obtain an interpretation of any triangular array of the form
\[
T(n,k)=(a_2 n + a_1 k + a_0)\,T(n-1,k)
+ (b_2 n + b_1 k + b_0)\,T(n-1,k-1).
\]
This lead to have an interpretation of $T(n,k)$ as an increasing tree.
Explicit formulas and structural properties are then derived through analytic differential equations.
In particular, the $r$-Whitney-Eulerian numbers and the cases where $b_2n+b_1k+b_0=1$ are obtained explicitly.
\noindent Applications include new interpretation formulas for the $r$-Eulerian numbers with generating functions. We also obtain full generating functions for the case $a_2=-a_1$ using this approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01005 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Triangular Arrays using context-free grammar Aubert, Voalaza Mahavily Romuald Randrianirina, Benjamin Combinatorics In this work, the Hao grammar $G=\{\, u\rightarrow u^{b_1+b_2+1} v^{a_1+a_2},\quad v\rightarrow u^{b_2}v^{a_2+1} \,\},$ together with the correspondence between grammars and combinatorial differential equations, is employed to obtain an interpretation of any triangular array of the form \[ T(n,k)=(a_2 n + a_1 k + a_0)\,T(n-1,k) + (b_2 n + b_1 k + b_0)\,T(n-1,k-1). \] This lead to have an interpretation of $T(n,k)$ as an increasing tree. Explicit formulas and structural properties are then derived through analytic differential equations. In particular, the $r$-Whitney-Eulerian numbers and the cases where $b_2n+b_1k+b_0=1$ are obtained explicitly. \noindent Applications include new interpretation formulas for the $r$-Eulerian numbers with generating functions. We also obtain full generating functions for the case $a_2=-a_1$ using this approach. |
| title | Triangular Arrays using context-free grammar |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.01005 |