Manon Stipulanti

Generalized Pascal triangle for binomial coefficients of words

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpinski 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.

Counting the number of non-zero coefficients in rows of generalized Pascal triangles

Abstract This paper is about counting the number of distinct (scattered) subwords occurring in a given word. More precisely, we consider the generalization of the Pascal triangle to binomial coefficients of words and the sequence ( S ( n ) ) n ≥ 0 counting the number of positive entries on each row. By introducing a convenient tree structure, we provide a recurrence relation for ( S ( n ) ) n ≥ 0 . This leads to a connection with the 2 -regular Stern–Brocot sequence and the sequence of denominators occurring in the Farey tree. Then we extend our construction to the Zeckendorf numeration system based on the Fibonacci sequence. Again our tree structure permits us to obtain recurrence relations for and the F -regularity of the corresponding sequence.