$q$-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings

TL;DR

This paper introduces new combinatorial interpretations of $q$-deformed rational numbers, extending previous work to all positive rationals and connecting them with Ostrowski numeration, posets, perfect matchings, and polytopes.

Contribution

It provides three novel enumerative interpretations of $q$-rationals applicable to all positive rationals, unifying various combinatorial objects and extending prior results.

Findings

New combinatorial models for $q$-rationals including Ostrowski numeration, posets, and perfect matchings

A formula for a $q$-analog of Markoff numbers

A geometric interpretation via integer points in polytopes

Abstract

We consider the $q$-deformation of rational numbers introduced recently by Morier-Genoud and Ovsienko. We propose three enumerative interpretations of these $q$-rationals: in terms of a new version of Ostrowski's numeration system for integers, in terms of order ideals of fence posets and in terms of perfect matchings of snake graphs. Contrary to previous results which are restricted to rational numbers greater than one, our interpretations work for all positive rational numbers and are based on a single combinatorial object for defining both the numerator and denominator. The proofs rest on order-preserving bijections between posets over these objects. We recover a formula for a $q$-analog of Markoff numbers. We also deduce a fourth interpretation given in terms of the integer points inside a polytope in $\mathbb{R}^k$ on both sides of a hyperplane where $k$ is the length of the continued fraction expansion.