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Updated: Jun 14, 2026
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Asymmetric Walkway: A Novel Behavioral Assay for Studying Asymmetric Locomotion
Published on: January 15, 2016
Staircase tableaux, the asymmetric exclusion process, and Askey-Wilson polynomials
Sylvie Corteel1, Lauren K Williams
1Laboratoire d'Informatique Algorithmique: Fondements et Applications, Centre National de la Recherche Scientifique et Université Paris Diderot-Paris 7, Case 7014, 75205, Paris Cedex 13, France.
Summary
We introduce staircase tableaux, combinatorial objects with cardinality 4(n)n!, connecting them to the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. This research provides formulas for ASEP stationary distributions and Askey-Wilson polynomial moments using these new objects.
Area of Science:
- Combinatorics
- Statistical Mechanics
- Orthogonal Polynomials
Background:
- The asymmetric exclusion process (ASEP) models particle dynamics on a lattice with open boundaries, relevant to traffic flow and protein synthesis.
- Askey-Wilson polynomials are crucial in the hierarchy of classical orthogonal polynomials, with prior work focusing on combinatorial formulas for their moments.
- Staircase tableaux are novel combinatorial objects introduced in this study.
Purpose of the Study:
- To introduce staircase tableaux as a new combinatorial object.
- To establish connections between staircase tableaux, the asymmetric exclusion process (ASEP), and Askey-Wilson polynomials.
- To derive new formulas for the stationary distribution of the ASEP and moments of Askey-Wilson polynomials.
Main Methods:
- Introduction of staircase tableaux as a combinatorial structure.
- Development of a formula for the stationary distribution of the general ASEP using staircase tableaux.
- Derivation of a formula for the moments of Askey-Wilson polynomials utilizing staircase tableaux.
Main Results:
- A formula for the stationary distribution of the asymmetric exclusion process (ASEP) with general parameters, expressed in terms of staircase tableaux.
- A formula for the moments of Askey-Wilson polynomials, also derived using staircase tableaux.
- The cardinality of staircase tableaux is established as 4(n)n!.
Conclusions:
- Staircase tableaux provide a unifying combinatorial framework for understanding the asymmetric exclusion process and Askey-Wilson polynomials.
- The derived formulas offer new insights into the statistical mechanics of the ASEP and the properties of Askey-Wilson polynomials.
- This work bridges combinatorics, statistical mechanics, and the theory of orthogonal polynomials.
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