A Combinatorial Bijection on di-sk Trees

A di-sk tree is a rooted binary whose nodes are labeled by $\oplus$ or $\ominus$, and no node has the same label as its right child. The trees in natural bijection with separable permutations. We construct combinatorial on proving two quintuples $(\mathrm{LMAX},\mathrm{LMIN},\mathrm{DESB},\mathsf{iar},\mathsf{comp})$ $(\mathrm{LMAX},\mathrm{LMIN},\mathrm{DESB},\mathsf{comp},\mathsf{iar})$ have distribution over Here for permutation $\pi$, $\mathrm{LMAX}(\pi)/\mathrm{LMIN}(\pi)$ set of values left-to-right maxima/minima $\pi$ $\mathrm{DESB}(\pi)$ descent bottoms while $\mathsf{comp}(\pi)$ $\mathsf{iar}(\pi)$ respectively number components length initial ascending run $\pi$. 
 Interestingly, our specializes to $312$-avoiding permutations, which provides (up classical Knuth–Richards bijection) an alternative approach result Rubey (2016) that asserts triples $(\mathrm{LMAX},\mathsf{iar},\mathsf{comp})$ $(\mathrm{LMAX},\mathsf{comp},\mathsf{iar})$ equidistributed $321$-avoiding Rubey's symmetric extension equidistribution due Adin–Bagno–Roichman, implies class permutations prescribed Schur positive. Some results various statistics concerning traversal presented end.