Noncommutative Catalan Numbers

Arkady Berenstein, Vladimir Retakh

Research output: Chapter in Book/Report/Conference proceedingChapter


The goal of this paper is to introduce and study noncommutative Catalan numbers Cn 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.

Original languageEnglish (US)
Title of host publicationTrends in Mathematics
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages21
StatePublished - 2021

Publication series

NameTrends in Mathematics
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


  • Catalan numbers
  • Laurent polynomials
  • Non-commuting variables


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