Pascal-like triangles: base 2 and beyond

The binomial coefficient (u,v) of two finite words u and v (on a finite alphabet) is the number of times the word v appears inside the word u as a subsequence (or, as a "scattered" subword). For instance, (abbabab,ab)=4. This concept naturally extends the classical binomial coefficients of integers, and has been widely studied for about thirty years (see, for instance, Simon and Sakarovitch). In this talk, I present the research lead since October 2015: I give the main ideas that give an extension of the Pascal triangle to base-2 expansions of integers and also give an overview of the results obtained so far, linked to this generalization. In the last part of this presentation, I extend the results to the Fibonacci setting.