Triangular Arrays using context-free grammar
Voalaza Mahavily Romuald Aubert, Benjamin Randrianirina

TL;DR
This paper uses context-free grammar and differential equations to interpret triangular arrays as increasing trees, deriving explicit formulas and applications to r-Whitney-Eulerian numbers.
Contribution
It introduces a novel grammar-based approach to interpret and analyze triangular arrays and related combinatorial numbers.
Findings
Derived explicit formulas for triangular arrays.
Provided new interpretations for r-Whitney-Eulerian numbers.
Obtained full generating functions for specific cases.
Abstract
In this work, the Hao grammar 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 as an increasing tree. Explicit formulas and structural properties are then derived through analytic differential equations. In particular, the -Whitney-Eulerian numbers and the cases where are obtained explicitly. \noindent Applications include new interpretation formulas for the -Eulerian numbers with generating functions. We also obtain full generating functions for the case using this approach.
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