  Recently, the second author studied an Eulerian statistic (called cover) in
the context of convex polytopes, and proved an equal joint distribution of
(cover,des) with (des,exc). In this paper, we present several direct bijective
proofs that cover is Eulerian, and examine its generalizations and their
Mahonian partners. We also present a quasi-symmetric function proof (suggested
by Michelle Wachs) of the above equal joint distribution.
