2024/03/10 by Armstrong, Drew
#05A30 (Primary) 17B22 #06A11 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2403.06318
For each pair of coprime integers a and b we have a rational q-Catalan number Cat(a,b)q=\binoma+baq/[a+b]q. It is known that this is a polynomial in q with nonnegative integer coefficients, but the nature of these coefficients is still mysterious. Our current understanding is based on the rational shuffle conjecture that was conjectured by Bergeron, Garsia, Leven and Xin in 2014 and proved by Mellit in 2016, based on earlier work with Carlsson. This theorem realizes Cat(a,b)q as the generating function for the statistic "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 Cat(a,c)q-Cat(a,b)q has nonnegative coefficients whenever gcd(a,b)=gcd(a,c)=1 and b