Explicit Formulas and Combinatorial Interpretation of Triangular Arrays
Voalaza Mahavily Romuald, Benjamin Randrianirina
TL;DR
The paper develops a unified lattice-path framework to study triangular arrays $T(n,k)$ defined by $T(n,k)=a_{n,k}T(n-1,k)+b_{n,k}T(n-1,k-1)$, focusing on affine forms for $a_{n,k}$ and $b_{n,k}$. It provides a weighted-path interpretation, a matrix-analytic perspective in the polynomial space, and explicit closed-form formulas for key cases, especially when $b_{n,k}=1$, linking to generalized Stirling numbers and $r$-Eulerian/Whitney-type numbers. The results yield explicit expressions for many combinatorial families and show how these arrays act as transition matrices between natural bases of polynomials, offering combinatorial interpretations and unifying several known generalizations. Collectively, the work supplies concrete tools for computing and interpreting generalized triangular arrays in enumerative combinatorics with potential applications to $r$-Eulerian numbers and related sequences.
Abstract
Using the lattice $\mathbb{N}\times\mathbb{N}$, we derive a general formula for the sequences $\big(T(n,k)\big)_{n,k\in \mathbb{N}}$ satysfying the recurence 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 general $T(n,k)$, 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 will analyze the case $b_{n,k}=1$, which frequently occurs in enumerative combinatorics. Applications include explicit formulas for the $r$-Eulerian numbers. We will write also the case where $b_{n,k}=1$, as a matrix of passage. \textbf{Keywords:} triangular recurrence, weighted paths, $r$-Eulerian numbers, combinatorial interpretation.