The goal of this paper is to introduce and study noncommutative Catalan numbersCn which belong to the free Laurent polynomial algebra Ln 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 Hn and introduce accompanying noncommutative binomial coefficients[InlineEquation not available: see fulltext.].
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics
- Catalan numbers
- Laurent polynomials
- Non-commuting variables