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

Jean-Christophe Aval, Sébastien Labbé

TL;DR

This work extends the Morier-Genoud–Ovsienko $q$-analogs of positive rationals to all ${x}>0$ by unifying three combinatorial interpretations: admissible sequences from even-length continued fractions, lower ideals of fence posets, and perfect matchings of snake graphs. It builds a network of explicit, structure-preserving bijections among these objects, enabling $q$-counting formulas for both numerators and denominators via a single fence poset and a single snake graph. It also derives a $q$-analog of Markoff numbers through area statistics on snake-graph matchings and reveals a convex polytope description for the admissible-sequence sets, connecting to polyhedral combinatorics and cluster-algebra frameworks. Overall, the paper provides a cohesive, all-positives-all-intervals framework that generalizes prior results (which were restricted to $x>1$) and links continued fractions, combinatorial posets, and geometric graphs in the study of $q$-rational numbers.

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.

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

TL;DR

This work extends the Morier-Genoud–Ovsienko -analogs of positive rationals to all by unifying three combinatorial interpretations: admissible sequences from even-length continued fractions, lower ideals of fence posets, and perfect matchings of snake graphs. It builds a network of explicit, structure-preserving bijections among these objects, enabling -counting formulas for both numerators and denominators via a single fence poset and a single snake graph. It also derives a -analog of Markoff numbers through area statistics on snake-graph matchings and reveals a convex polytope description for the admissible-sequence sets, connecting to polyhedral combinatorics and cluster-algebra frameworks. Overall, the paper provides a cohesive, all-positives-all-intervals framework that generalizes prior results (which were restricted to ) and links continued fractions, combinatorial posets, and geometric graphs in the study of -rational numbers.

Abstract

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

Paper Structure

This paper contains 23 sections, 38 theorems, 164 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. Continued fractions
  3. Continued fraction expansions of rational numbers
  4. Continued fraction convergents in matrix form
  5. Binary encodings of rational numbers
  6. Two involutions on rational numbers
  7. Morier-Genoud--Ovsienko's q-analog of rational numbers
  8. A bijection from admissible sequences to integers in an interval
  9. A bijection from fence poset order ideals to admissible sequences
  10. From rational numbers to fence posets
  11. Fence poset order ideals
  12. A bijection from fence poset order ideals to admissible sequences
  13. Proof of Theorem \ref{['thm:1-norm-statistics']} and Theorem \ref{['thm:nice-formula-for-order-ideals']}
  14. A first application: counting order ideals with cardinality statistics
  15. A second application: counting admissible sequences with 1-norm statistics
  16. ...and 8 more sections

Key Result

Theorem A

Let $k\geq1$ be an integer and $a=[a_0; a_1, \dots, a_{k-1}]$ be the even or odd-length continued fraction expansion of a positive rational number. Let ${\mathcal{Z}}(a)=\mathbb{Z}\,\cap\,[0,r_{k})$ if $k$ is odd or ${\mathcal{Z}}(a)=\mathbb{Z}\cap[r_{k-1}-r_{k},r_{k-1})$ if $k$ is even. Every integ where $(b_i)_{0\leq i\leq k-1}$ is an admissible sequence for $a$.

Figures (21)

  • Figure 1: The big picture of bijections presented in this article.
  • Figure 2: The admissible sequences ${\mathcal{B}}(\frac{4}{5})$ for $\frac{4}{5}=[0;1,3,1]$, the order ideals ${\mathcal{J}}(\frac{4}{5})$ of the fence poset ${\mathcal{F}}(\frac{4}{5})$, and the set of perfect matchings ${\mathcal{M}}(\frac{4}{5})$ of the snake graph ${\mathcal{G}}(\frac{4}{5})$ are isomorphic posets. The poset is partitioned into two disjoint poset intervals whose rank polynomials are the numerator and denominator of $[4/5]_q$ up to a scalar $q^{-1}$.
  • Figure 3: The fence posets $F(\mathtt{1}\mathtt{1}\mathtt{1}\mathtt{0}\mathtt{0}\mathtt{1})$ and $F(\mathtt{00011011100})$.
  • Figure 4: The map ${\mathcal{F}}$ maps the Stern-Brocot tree of positive rational numbers on the binary tree of fence posets ordered by the prefix relation.
  • Figure 5: The order ideal $\{0,1,6,7,8,9,13,14\}$ of cardinality $8$ in the distributive lattice ${\mathcal{J}}^\bullet(399/121)={\mathcal{J}}^\bullet([3,3,2,1,3,3])$ is shown with dark vertices. The elements $y_i$ of the poset are identified with their indices $i$.
  • ...and 16 more figures

Theorems & Definitions (87)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Corollary E
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 77 more