Journal of Algebraic Combinatorics

, Volume 36, Issue 3, pp 333–354 | Cite as

Double Catalan monoids

  • Volodymyr Mazorchuk
  • Benjamin Steinberg


In this paper, we define and study what we call the double Catalan monoid. This monoid is the image of a natural map from the 0-Hecke monoid to the monoid of binary relations. We show that the double Catalan monoid provides an algebraization of the (combinatorial) set of 4321-avoiding permutations and relate its combinatorics to various off-shoots of both the combinatorics of Catalan numbers and the combinatorics of permutations. In particular, we give an algebraic interpretation of the first derivative of the Kreweras involution on Dyck paths, of 4321-avoiding involutions and of recent results of Barnabei et al. on admissible pairs of Dyck paths. We compute a presentation and determine the minimal dimension of an effective representation for the double Catalan monoid. We also determine the minimal dimension of an effective representation for the 0-Hecke monoid.


Catalan monoid Presentation Pattern avoiding permutation Effective representation 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  3. 3.Department of MathematicsCity College of New YorkNew YorkUSA

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