Noncommutative Catalan numbers
The goal of my talk (based on joint work with Vladimir Retakh) is to introduce and study noncommutative versions of Catalan numbers which belong to the free Laurent polynomial algebra L_n in n generators.
Our noncommutative Catalan numbers C_n admit interesting (commutative and noncommutative) specializations, one of them related to Garsia-Haiman (q,t)-versions, another -- to solving noncommutative
quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices H_n and introduce two kinds of noncommutative binomial coefficients which are instrumental
in computing the inverse of H_n and in other combinatorial identities involving C_n.