Triangular Arrays using context-free grammar

TL;DR

The paper develops a grammar-based framework that translates triangular recurrences of the form $T(n,k)=(a_2 n+a_1 k+a_0)T(n-1,k)+(b_2 n+b_1 k+b_0)T(n-1,k-1)$ into context-free grammars and associated combinatorial differential equations. By pairing Hao's grammar with exponential generating functions, it derives explicit formulas and structural properties, including $r$-Whitney–Eulerian numbers and special-case simplifications when $b_2n+b_1k+b_0=1$. It also connects these recurrences to refined counting via type-(E) grammars and provides a generating-function approach to descents in Stirling $r$-permutations, notably for $r=2$. The results offer a unifying, grammar-driven method for interpreting and solving triangular arrays in combinatorics, with explicit formulas and generating functions that extend classical Eulerian-type numbers.

Abstract

In this work, the grammar of Hao \[ G=\{\, u\rightarrow u^{b_1+b_2+1} v^{a_1+a_2},\quad v\rightarrow u^{b_2}v^{a_2+1} \,\}, \] together with the correspondence between grammars and Combinatorial Differential Equations, is employed to obtain an interpretation of any triangular array of the form \[ T(n,k)=(a_2 n + a_1 k + a_0)\,T(n-1,k) + (b_2 n + b_1 k + b_0)\,T(n-1,k-1). \] 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. Applications include new interpretation formulas for the $r$-Eulerian numbers with generating function for a special case. Keywords: triangular recurrence, formal grammar, Combinatorial operators, differential equations,$r$-Eulerian, combinatorial interpretation, $r$-Whitney--Eulerian.

Figures (10)

• Figure 1: Integral equation
• Figure 2: System of integral equations
• Figure 3: $\mathcal{D}^n(u^{m-r}v^r)$-structure for $n=0$
• Figure 4: $\mathcal{D}^n(u^{m-r}v^r)$-structures for $n=1$
• Figure 5: $\mathcal{D}^n(u^{m-r}v^r)$-structures for $n=2$

Theorems & Definitions (27)

• Definition 1.1: Weighted set
• Definition 1.2
• Definition 1.3: Randrianirina2000
• Definition 1.4
• Definition 1.5: randrianirina2000combinatoire
• Remark 1.1
• Definition 1.6
• Proposition 2.1: hao2015context
• Proposition 2.2
• Theorem 2.1