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The Catalan's triangle system, the Catalan's trapezoids and (q,2)--Fock space

Yungang Lu

Abstract

We provide an explicit formulation for the solution to the Catalan's triangle system using Catalan's trapezoids and a specified boundary condition. Additionally, we study this system with various boundary conditions obtained by utilizing different types of Fock spaces.

The Catalan's triangle system, the Catalan's trapezoids and (q,2)--Fock space

Abstract

We provide an explicit formulation for the solution to the Catalan's triangle system using Catalan's trapezoids and a specified boundary condition. Additionally, we study this system with various boundary conditions obtained by utilizing different types of Fock spaces.
Paper Structure (9 sections, 17 theorems, 163 equations)

This paper contains 9 sections, 17 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. The Catalan numbers, Catalan's triangle and non--crossing pair partition
  3. Reformulation of the system \ref{['DCT01y']} and the Catalan's triangle system
  4. Pair partition, non--crossing pair partition and some their properties
  5. Non--crossing pair partition and the Catalan's triangle system
  6. Solving the Catalan's triangle system \ref{['DCT25a']}
  7. $(q,2)-$Fock space, its involved pair partitions and connection to the system \ref{['DCT25a']}
  8. The usual Fock space and pair partitions
  9. The $(q,2)-$Fock space and some of its elementary properties

Key Result

Proposition 2.2

For any $n\in\mathbb{N}^*$ and $\{(l_h,r_h)\}_{h=1}^n\in PP(2n)$, In particular, $l_1=1$ and $l_n<2n$.

Theorems & Definitions (43)

  • Remark 1.1
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6
  • ...and 33 more