Noncommutative Catalan numbers

TL;DR

This paper introduces noncommutative Catalan numbers within a free Laurent polynomial algebra, exploring their specializations, positivity properties, and related noncommutative binomial coefficients, expanding combinatorial and algebraic understanding.

Contribution

It defines noncommutative Catalan numbers, studies their specializations, and proves total positivity of associated Hankel matrices, providing new algebraic tools and insights.

Findings

Noncommutative Catalan numbers are introduced and characterized.

Total positivity of noncommutative Hankel matrices is established.

Noncommutative binomial coefficients are defined and analyzed.

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 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.