Noncommutative Catalan Numbers

Arkady Berenstein, Vladimir Retakh

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.].

Original languageEnglish (US)
Pages (from-to)527-547
Number of pages21
JournalAnnals of Combinatorics
Issue number3-4
StatePublished - Nov 1 2019

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics


  • Catalan numbers
  • Laurent polynomials
  • Non-commuting variables


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