Lattice Points and Rational q-Catalan Numbers

Abstract For each pair of coprime integers a and b , we have the rational q -Catalan number $$\operatorname {Cat}(a,b)_q=\bigl [\hspace{-1.5pt}{\begin{smallmatrix}{a+b}\\ {a}\end{smallmatrix}}\hspace{-1.0pt}\bigr ]_q/[a+b]_q.$$ Cat ( a , b ) q = [ a + b a ] q / [ a + b ] q . It is known that this is a polynomial in q with non-negative integer coefficients, but the nature of these coefficients is still mysterious. Our current understanding is based on the rational shuffle conjecture of Bergeron, Garsia, Leven, and Xin from 2014, which was proved by Mellit in 2016, building on earlier work with Carlsson. This theorem realizes $$\operatorname {Cat}(a,b)_q$$ Cat ( a , b ) q as the generating function for the statistic “ $$\operatorname {area}-\operatorname {dinv}+(a-1)(b-1)/2$$ area - dinv + ( a - 1 ) ( b - 1 ) / 2 ” defined on rational Dyck paths. However, this statistic is difficult to work with and leaves some phenomena unexplained. For example, it does not prove the conjecture that the difference $$\operatorname {Cat}(a,c)_q-\operatorname {Cat}(a,b)_q$$ Cat ( a , c ) q - Cat ( a , b ) q has non-negative coefficients whenever $${\textrm{gcd}}(a,b)={\textrm{gcd}}(a,c)=1$$ gcd ( a , b ) = gcd ( a , c ) = 1 and $$b b < c .