  We solve two problems regarding the enumeration of lattice paths in
$\mathbb{Z}^2$ with steps $(1,1)$ and $(1,-1)$ with respect to the major index,
defined as the sum of the positions of the valleys, and to the number of
certain crossings. The first problem considers crossings of a single path with
a fixed horizontal line. The second one counts pairs of paths with respect to
the number of times they cross each other. Our proofs introduce lattice path
bijections with convenient visual descriptions, and the answers are given by
remarkably simple formulas involving $q$-binomial coefficients.
