English

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

Combinatorics 2025-11-17 v1 Number Theory

Abstract

We consider the qq-deformation of rational numbers introduced recently by Morier-Genoud and Ovsienko. We propose three enumerative interpretations of these qq-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 qq-analog of Markoff numbers. We also deduce a fourth interpretation given in terms of the integer points inside a polytope in Rk\mathbb{R}^k on both sides of a hyperplane where kk is the length of the continued fraction expansion.

Keywords

Cite

@article{arxiv.2511.11290,
  title  = {$q$-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings},
  author = {Jean-Christophe Aval and Sébastien Labbé},
  journal= {arXiv preprint arXiv:2511.11290},
  year   = {2025}
}

Comments

48 pages, 21 figures, 1 table