Hyperbinary partitions and q -deformed rationals

A hyperbinary partition of the nonnegative integer n is a partition where every part is a power of $2$ and every power of $2$ appears at most twice. We give three applications of the length generating function for such partitions, denoted by hq(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\,hq(n-1)/hq(n) where $r/s$ occurs as the n th 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, Z) . We express the entries of these products in terms of the polynomials hq(n) .