# Hyperbinary partitions and q-deformed rationals

## TL;DR

This paper explores hyperbinary partitions and their q-analogues of rationals, revealing new formulas, poset structures, and matrix representations that connect these concepts in combinatorics and algebra.

## Contribution

It introduces a novel connection between hyperbinary partitions, q-analogues of rationals, and poset structures, providing new formulas and matrix representations.

## Key findings

- The q-analogue of a rational number relates to hyperbinary partitions via a specific formula.
- Fence posets are isomorphic to lattices of hyperbinary partitions.
- Matrix entries for q-analogues can be expressed using hyperbinary partition polynomials.

## Abstract

A hyperbinary partition of the nonnegative integer n is a partition where every part is a power of 2 and every part appears at most twice. We give three applications of the length generating function for such partitions, denoted by h_q(n). Morier-Genoud and Ovsienko defined the q-analogue of a rational number [r/s]_q in various ways, most of which depend directly or indirectly on the continued fraction expansion of r/s. As our first application we show that [r/s]_q = q h_q(n-1)/h_q(n) where r/s occurs as the nth entry in the Calkin-Wilf enumeration of the non-negative rationals. Next we consider fence posets which are those which can be obtained from a sequence of chains by alternately pasting together maxima and minima. For every n we show there is a fence poset F(n) whose lattice of order ideals is isomorphic to the poset of hyperbinary partitions of n ordered by refinement. For our last application, Morier-Genoud and Ovsienko also showed that [r/s]_q can be computed by taking products of certain matrices which are q-analogues of the standard generators for the special linear group SL(2,R). We express the entries of these products in terms of the polynomials h_q(n).