Extensions of the Pascal Triangle to Words, and Related Counting Problems

The Pascal triangle and the corresponding Sierpiński fractal are fairly well-studied mathematical objects, which both exhibit connections with many different scientific areas. The first is made of binomial coefficients of integers that notably appear in combinatorics to tackle counting problems (for instance, they provide the number of possible ways to choose a given amount of elements from a set of elements). There exist multiple generalizations of those binomial coefficients. In this text, we focus on binomial coefficients of words, which count scattered subwords. The red thread of this thesis is precisely the combination of the Pascal triangle and binomial coefficients of words. The first part is dedicated to extensions of the Pascal triangle to various sets of words (languages) associated with different numeration systems. We transport the existing link between the Pascal triangle and the Sierpiński gasket to this wider setting. The second part is concerned with particular sequences extracted from generalized Pascal triangles. They count non-zeroes binomial coefficients on each row of a given Pascal-like triangle. We study their regularity and their automaticity with respect to different numeration systems. In the third and last part, we establish the asymptotics of the summatory functions of the sequences considered previously. The most important feature of this part might not necessarily be the result itself, but the underlying new method to achieve it.