Oriented posets, Rank Matrices and q-deformed Markov Numbers

TL;DR

This paper introduces oriented posets and rank matrices, linking them through matrix multiplication, and applies this framework to model q-deformed Markov numbers, resolving a conjecture and deriving new identities.

Contribution

It develops a novel combinatorial framework using oriented posets and rank matrices, providing a new model for q-deformed Markov numbers and resolving an existing conjecture.

Findings

Linked chains to fence posets via matrix multiplication

Provided a combinatorial model for q-deformed Markov numbers

Resolved a conjecture of Leclere and Morier-Genoud

Abstract

We define oriented posets with correpsonding rank matrices, where linking two posets by an edge corresponds to matrix multiplication. In particular, linking chains via this method gives us fence posets, and taking traces gives us circular fence posets. As an application, we give a combinatorial model for $q$-deformed Markov numbers. We also resolve a conjecture of Leclere and Morier-Genoud and give several identities between circular rank polynomials.