On a conjecture concerning two-stack sortable permutations

The two-stack sortable permutations of length $n$ are known to be equinumerous with permutations of length $n$ avoiding the dashed patterns $3\text{-}1\text{-}4\text{-}2$ and $2\text{-}41\text{-}3$. In this paper, we confirm an intriguing conjecture posed by Claesson, Kitaev, and Steingrímsson, which asserts that the quadruple $(\text{comp},\text{asc},\text{ldr},\mathrm{a}x)$ is equidistributed on these two permutation classes. Here $\text{comp}$ and $\text{asc}$ count the number of components and ascents, respectively, $\text{ldr}$ denotes the length of the initial decreasing run, and $\mathrm{a}x$ denotes the number of right-to-left maxima. This is accomplished by constructing bijections between two-stack sortable permutations and $\beta (1,0)$-trees.