A combinatorial bijection on di-sk trees

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