Lattice Points and Rational q-Catalan Numbers

For each pair of coprime integers a and b one defines the "rational q-Catalan number" Catq(a,b)=[-1.5pt smallmatrixa-1+b\-1smallmatrix-1pt]q/[a]q. It is known that this is a polynomial in q with nonnegative integer coefficients, but this phenomenon is mysterious. Despite recent progress in the understanding of these polynomials and their two-variable $q,t$-analogues, we still lack a simple combinatorial interpretation of the coefficients. The current paper builds on a conjecture of Paul Johnson relating q-Catalan numbers to lattice points. The main idea of this approach is to fix a and express everything in terms of the weight lattice of type Aₐ₋₁. For a given a we construct a family of (a-2)φ(a)+1 polynomials called "q-Catalan germs" and for each integer b coprime to a we express Cat(a,b)q in terms of germs. We conjecture that the germs have nonnegative coefficients and we show that this nonnegativity conjecture is implied by a stronger conjecture about "ribbon partitions" of certain subposets of Young's lattice.