Generalized Pascal triangle for binomial coefficients of words : an overview

We introduce a generalization of 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. Similarly to the Sierpiński gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1] × [0, 1] associated with this extended Pascal triangle modulo a prime p. Then we create a new sequence from this extended Pascal triangle that counts, on each row of this triangle, the number of positive binomial coefficients. We study some properties of this sequence. To be precise, we investigate some properties regarding the regularity of the sequence. To extend our work, we construct a Pascal triangle using the Fibonacci representations of all the nonnegative integers and we define the corresponding sequence of which we study the regularity. This regularity is an extension of the classical k-regularity of sequences.