  Recently O. Pechenik studied the cyclic sieving of increasing tableaux of
shape $2\times n$, and obtained a polynomial on the major index of these
tableaux, which is a $q$-analogue of refined small Schr\"{o}der numbers. We
define row-increasing tableaux and study the major index and amajor index of
row-increasing tableaux of shape $2 \times n$. The resulting polynomials are
both $q$-analogues of refined large Schr\"{o}der numbers. For both results we
give bijective proofs.
