Bijection Between Increasing Binary Trees and Rook Placements on Double Staircases

In this paper, we shall construct a bijection between rook placements on double staircases (introduced by Josuat-Vergès in 2017) and increasing binary trees. We introduce two subclasses of staircases, which call left right-aligned placements. show that their enumeration, while keeping track certain statistic, gives the $\gamma$-vectors Eulerian polynomials. conclude with discussion different fits very well our main bijection, another generalising bijection. Our is special case due to Tewari (2019).