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Description

The goal of my talk (based on joint work with Vladimir Retakh) is to introduce noncommutative analogs 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 its positive factorizations, and other combinatorial identities involving \(C_n\).

If time permits, I will explain the relationship of the \(C_n\) with the noncommutative Laurent Phenomenon, which was previously established for Kontsevich rank 2 recursions and all marked surfaces.