Séminaire Lotharingien de Combinatoire, 95B.63 (2026), 12 pp.
Thomas McConville, James Propp and Bruce E. Sagan
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 part 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
= qhq(n-1)/hq(n)
where r/s occurs as the nth entry in the
Calkin-Wilf enumeration of the nonnegative rationals. Next we consider
fence posets which are those obtained from a sequence of chains by
alternately pasting together maxima and minima. For every n we
show there is a fence poset 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. We express the entries of these products in terms of the
polynomials hq(n).
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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