  Given a coprime pair $(m,n)$ of positive integers, rational Catalan numbers
$\frac{1}{m+n} \binom{m+n}{m,n}$ counts two combinatorial objects:rational
$(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go
below the diagonal; $(m,n)$-cores are partitions with no hook length equal to
$m$ or $n$.Anderson established a bijection between $(m,n)$-Dyck paths and
$(m,n)$-cores. We define a new transformation, called rank complement, on
rational Dyck paths. We show that rank complement corresponds to conjugation of
$(m,n)$-cores under Anderson's bijection. This leads to: i) a new approach to
characterizing $n$-cores; ii) a simple approach for counting the number of
self-conjugate $(m,n)$-cores; iii) a proof of the equivalence of two
conjectured combinatorial sum formulas, one over rational $(m,n)$-Dyck paths
and the other over $(m,n)$-cores, for rational Catalan polynomials.
