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.

中文翻译：

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非可交换加泰罗尼亚数

本文的目标是引进和研究非交换Catalan数\（C_N \）属于免费洛朗多项式代数\（\ mathcal {L} _n \）在ñ发电机。我们的非可交换数允许有趣的（可交换和非可交换）专业化，其中之一与Garsia–Haiman（q，  t）版本有关，另一种与非交换二次方程有关。我们还建立了相应的（非交换）Hankel矩阵\（H_n \）的总正性，并介绍了伴随的非交换二项式系数。