Cores with distinct parts and bigraded Fibonacci numbers

The notion of (a,b)-cores is closely related to rational (a,b)-Dyck paths via the bijection due to J. Anderson, and thus the number of (a,a+1)-cores is given by the Catalan number Ca. Recent research shows that (a,a+1)-cores with distinct parts are enumerated by another important sequence- Fibonacci numbers Fa. In this paper, we consider the abacus description of (a,b)-cores to introduce the natural grading and generalize this result to (a,as+1)-cores. We also use the bijection with Dyck paths to count the number of (2k−1,2k+1)-cores with distinct parts. We give a second grading to Fibonacci numbers, induced by the bigraded Catalan sequence Ca,b(q,t).