Bi-banded Paths, a Bijection and the Narayana Numbers
Judy-anne Osborn
TL;DR
This paper establishes a bijection between bi-banded paths and peak-counting paths, providing a new combinatorial interpretation of Narayana numbers through weight polynomials of bi-banded Dyck paths, linked to the TASEP process.
Contribution
It introduces a novel bijection between bi-banded and peak-counting paths, offering a new perspective on Narayana numbers in the context of lattice path enumeration.
Findings
Narayana numbers are interpreted as coefficients of weight polynomials for bi-banded Dyck paths.
A bijection between bi-banded paths and peak-counting paths is constructed.
The results connect combinatorial path enumeration with stochastic process models.
Abstract
We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating bi-banded Dyck paths, which class of paths has arisen naturally in previous literature in a solution of the stationary state of the `TASEP' stochastic process.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Random Matrices and Applications · Stochastic processes and statistical mechanics