  The goal of this paper is to introduce and study noncommutative Catalan
numbers $C_n$ which belong to the free Laurent polynomial algebra in $n$
generators. Our noncommutative numbers 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 $H_m$ and introduce accompanying noncommutative binomial coefficients.

Quantum Algebra;Combinatorics;Representation Theory