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

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.