Généralisations du coefficient binomial de mots

Binomial coefficients of natural numbers are ubiquitous in mathematics and have a rich and fascinating history. Among the various generalizations of these coefficients, we focus on the one that extends them to words over a finite alphabet A: given two words u and v, their binomial coefficient counts the number of times v appears as a subsequence of u. Recently, we introduced a q-deformation of these coefficients, which provides additional information about those subsequences. We have also generalized connected concepts such as Parikh matrices. In another direction, we are currently exploring binomial coefficients defined over the free group F(A) and their connections to binomial coefficients of integers. This poster aims to provide an overview of the current state of research in this area, highlighting both the achievements and the future directions.