The notion of (a,b)-cores is closely related to rational (a,b) Dyck paths
due to Anderson's bijection, and thus the number of (a,a+1)-cores is given by
the Catalan number Caβ. Recent research shows that (a,a+1) cores with
distinct parts are enumerated by another important sequence- Fibonacci numbers
Faβ. 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
(2kβ1,2k+1)-cores with distinct parts. We give a second grading to Fibonacci
numbers, induced by bigraded Catalan sequence Ca,bβ(q,t)