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

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 Sierpiński 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] × [0, 1] associated with the latter generalization of the Pascal triangle modulo a prime number.