Abstract
The goal of this paper is to introduce and study noncommutative Catalan numbers$$C_n$$ which belong to the free Laurent polynomial algebra $$\mathcal {L}_n$$ 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 $$H_n$$ and introduce accompanying noncommutative binomial coefficients.
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