  We give a direct combinatorial proof of the equidistribution of two pairs of
permutation statistics, (des, aid) and (lec, inv), which have been previously
shown to have the same joint distribution as (exc, maj), the major index and
the number of excedances of a permutation. Moreover, the triple (pix, lec, inv)
was shown to have the same distribution as (fix, exc, maj), where fix is the
number of fixed points of a permutation. We define a new statistic aix so that
our bijection maps (pix, lec, inv) to (aix, des, aid). We also find an Eulerian
partner das for a Mahonian statistic mix defined using mesh patterns, so that
(das, mix) is equidistributed with (des, inv).
