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Catalan numbers: from FC elements to classical diagram algebras

Sadek Al Harbat

Abstract

Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-type Coxeter group. Using only the settings of their canonical form, we recount $W^c(A_n)$ by the recurrence that is taken as a definition of the Catalan number $C_{n+1}$ and we find the Narayana numbers as well as the Catalan triangle via suitable set partitions of $W^c(A_n)$. We determine the unique bijection between $W^c(A_n)$ and the set of non-crossing diagrams of $n+1$ strings that respects the diagrammatic multiplication by concatenation in the $A_n$-type Temperley-Lieb algebra, along with the two algorithms implementing this bijection and its inverse.

Catalan numbers: from FC elements to classical diagram algebras

Abstract

Let be the set of fully commutative elements in the -type Coxeter group. Using only the settings of their canonical form, we recount by the recurrence that is taken as a definition of the Catalan number and we find the Narayana numbers as well as the Catalan triangle via suitable set partitions of . We determine the unique bijection between and the set of non-crossing diagrams of strings that respects the diagrammatic multiplication by concatenation in the -type Temperley-Lieb algebra, along with the two algorithms implementing this bijection and its inverse.
Paper Structure (30 sections, 21 theorems, 71 equations, 7 figures)

This paper contains 30 sections, 21 theorems, 71 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Canonical form and basic bijections
  3. A canonical form for FC elements
  4. Catalania: Dyck paths and ballots
  5. New set-partitions: Catalan numbers coming directly from FC elements
  6. FC elements counted by the inductive definition of Catalan numbers.
  7. Sub-partitions (one parameter partitions of $C_{n+1}$): Narayana numbers
  8. Sub-partitions (one parameter partitions of $C_{n+1}$): Catalan triangle
  9. Double partitions (two parameters partitions of $C_{n+1}$) and more
  10. "The" explicit bijection between non-crossing diagrams and FC elements
  11. The definitions
  12. The monomial basis
  13. Non-crossing diagrams
  14. Multiplication of non-crossing diagrams
  15. Aside: another bijection between FC elements and diagrams.
  16. ...and 15 more sections

Key Result

Theorem 2.2

St Let $n$ be a positive integer, then $A^c_n$ is the set of elements of the form: We call size of $w$ the integer $p$ ( with $0$ as the size of the identity). We call such pairs standard $p$-pairs.

Figures (7)

  • Figure 1: $D_1$ (left), $\ D_2$ (right)
  • Figure 2: $D_1 \ast D_2$ (left), $\ D_2 \ast D_1$ (right)
  • Figure 3: Diagram $E_i$
  • Figure 4: Counterexample
  • Figure 5: $\mathcal{D}( [ i_1, j_1 ])$
  • ...and 2 more figures

Theorems & Definitions (39)

  • Example 1.1
  • Definition 2.1
  • Theorem 2.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • ...and 29 more