  We present refined enumeration formulas for lattice paths in $\mathbb{Z}^2$
with two kinds of steps, by keeping track of the number of descents (i.e.,
turns in a given direction), the major index (i.e., the sum of the positions of
the descents), and the number of crossings. One formula considers crossings
between a path and a fixed line; the other considers crossings between two
paths. Building on the first paper of the series, which used lattice path
bijections to give the enumeration with respect to major index and crossings,
we obtain a refinement that keeps track of the number of descents. The proof is
based on new bijections which rely on certain two-rowed arrays that were
introduced by Krattenthaler.

Combinatorics