Noncommutative Catalan numbers
Abstract
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 GarsiaHaiman $(q,t)$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.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 DOI:
 10.48550/arXiv.1708.03316
 arXiv:
 arXiv:1708.03316
 Bibcode:
 2017arXiv170803316B
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Combinatorics;
 Mathematics  Representation Theory
 EPrint:
 12 pages AM LaTex, a picture and proof of Lemma 3.6 are added, misprints corrected