A way to extend Pascal's triangle to words

Pascal's triangle and the corresponding Sierpiński's triangle are well-studied objects and have connections with different areas in science. The main ingredient of this presentation is the link between them. I will first recall it and then exploit it to present a way of extending both objects to the area of combinatorics on words. Combinatorics on words is a relatively new domain of discrete mathematics, which focuses on the study of words and formal languages. In this context, a finite word is simply a finite sequence of letters, or symbols, that belong to a finite set called the alphabet. For instance, 01101 and 01 are two finite (binary) words over the (binary) alphabet {0,1}. A language is a set of words. For instance, we let {0,1}* denote the set of all finite words over {0,1}. The binomial coefficient of two finite words u and v over some alphabet is the number of occurrences of v as a subsequence of u. For example, the binomial coefficient of 01101 and 01 is 4. This concept, which generalizes binomial coefficients of integers, has been widely studied for the last thirty years or so. Knowing the definition of the Pascal's triangle with binomial coefficients of integers, its extension to binomial coefficients of words seems somewhat natural.