Generalized Schröder paths and Young tableaux with skew shapes

Xiaomei Chen

Published 2020-02-05Version 1

A generalized Schr\"{o}der path is a lattice path with steps (1,0), (1,1) and (0,1), and never goes above the diagonal line $y=x$. In this paper, we firstly give 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. 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 a skew shape is obtained.