$q$-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908653356843008 |
|---|---|
| author | Aval, Jean-Christophe Labbé, Sébastien |
| author_facet | Aval, Jean-Christophe Labbé, Sébastien |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11290 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $q$-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings Aval, Jean-Christophe Labbé, Sébastien Combinatorics Number Theory 05A15, 05A30, 11A63 (Primary) 05A19, 05C10, 06A07, 11A55, 11J06, 52A20, 68R15 (Secondary) 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. |
| title | $q$-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings |
| topic | Combinatorics Number Theory 05A15, 05A30, 11A63 (Primary) 05A19, 05C10, 06A07, 11A55, 11J06, 52A20, 68R15 (Secondary) |
| url | https://arxiv.org/abs/2511.11290 |