Enumeration of row-increasing tableaux of two-row skew shapes

In this paper, we firstly extend a result of Bonin, Shapiro and Simion by giving the distribution of the major index over generalized Schr\"{o}der paths. Then by providing a bijection between generalized Schr\"{o}der paths and row-increasing tableaux of skew shapes with two rows, we obtain the distribution of the major index and the amajor index over these tableaux, which extends a result of Du, Fan and Zhao. We also generalize a result of Pechenik and give the distribution of the major index over increasing tableaux of skew shapes with two rows. Especially, a bijection from row-increasing tableaux with shape $(n,m)$ and maximal value $n+m-k$ to standard Young tableaux with shape $((n-k+1,m-k+1,1^k)/(1^2))$ is obtained.