Hyperbinary partitions and q -deformed rationals

Abstract 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/sq 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/sq=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/sq 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).