An area-bounce exchanging bijection on a large subset of Dyck paths

It is a longstanding open problem to find a bijection exchanging area and bounce statistics on Dyck paths. We settle this problem for an exponentially large subset of Dyck paths via an explicit bijection. Moreover, we prove that this bijection is natural by showing that it maps what we call bounce-minimal paths to area-minimal paths. As a consequence of the proof ideas, we show combinatorially that a path with area $a$ and bounce $b$ exists if and only if a path with area $b$ and bounce $a$ exists. We finally show that the number of distinct values of the sum of the area and bounce statistics is the number of nonzero coefficients in Johnson’s $q$-Bell polynomial.