Convergence of Pascal-Like Triangles in Parry-Bertrand Numeration Systems

TL;DR

This paper explores the convergence properties of Pascal-like triangles constructed from binomial coefficients of finite words within Parry-Bertrand numeration systems, revealing fractal structures analogous to the Sierpiński gasket.

Contribution

It introduces a new generalization of Pascal triangles based on finite words in Parry-Bertrand systems and studies their limit sets in relation to fractal geometry.

Findings

The generalized Pascal triangles converge to fractal sets similar to the Sierpiński gasket.

The subset of [0,1]×[0,1] associated with these triangles is characterized.

The structure depends on the prime modulus used in the generalization.

Abstract

We pursue the investigation of generalizations of the Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a finite word appears as a subsequence of another finite word. The finite words occurring in this paper belong to the language of a Parry numeration system satisfying the Bertrand property, i.e., we can add or remove trailing zeroes to valid representations. It is a folklore fact that the Sierpi\'{n}ski gasket is the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from the classical Pascal triangle modulo $2$. In a similar way, we describe and study the subset of $[0, 1] \times [0, 1]$ associated with the latter generalization of the Pascal triangle modulo a prime number.