Oriented posets, Rank Matrices and q-deformed Markov Numbers

TL;DR

Bridges poset combinatorics and $q$-analogs by introducing oriented posets with rank matrices that multiply under edge connections; uses this framework to model fence and circular fence posets and to develop a combinatorial interpretation of $q$-deformed Markov numbers via circular fence rank polynomials; proves identities and partial unimodality results and resolves a standing conjecture on unimodality for circular rank polynomials; suggests broader applications to cluster algebras and other $q$-deformations of algebraic structures.

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.

Figures (9)

• Figure 1: Posets $F(2,1,1,3)$ (left) and $\overline{F}(2,1,1,3)$ (right).
• Figure 2: Posets $F^\updownarrow(2,1,1,3)$ (left) and $\overline{F}^\updownarrow(2,1,1,3)=\overline{F}(1,1,3,2)$ (right).
• Figure 3: $\overrightarrow{U_3}\searrow\overrightarrow{D_3}$ gives us the fence poset for $(3,4)$.
• Figure 4: The posets $\overrightarrow{U_3} \nearrow \overrightarrow{D_3}$ (left) and $\circlearrowleft\! (\overrightarrow{U_3} \nearrow \overrightarrow{D_3} )$ (right).
• Figure 5: Illustration of (Id 0.1) at $s=4$, $k=3$

Theorems & Definitions (43)

• Example 1.1
• Example 1.2
• Theorem 1.3: ourpaper
• Conjecture 1.4: ourpaper
• Conjecture 1.5: leclere, Conjecture 3.12
• Theorem 1: \ref{['thm:3']}
• Theorem 2: \ref{['thm:markov']}
• Example 2.1
• Example 2.2
• Lemma 2.3: leclere