Double Catalan monoids

TL;DR

This paper introduces the double Catalan monoid, an algebraic structure linked to 4321-avoiding permutations, and explores its combinatorial properties, representations, and connections to Catalan number-related structures.

Contribution

It defines the double Catalan monoid, relates it to permutation patterns and Catalan combinatorics, and computes minimal effective representations for it and the 0-Hecke monoid.

Findings

Provides a presentation of the double Catalan monoid

Determines the minimal dimension of effective representations

Establishes algebraic connections to Dyck paths and permutation classes

Abstract

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 {\em 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.