A combinatorial bijection on di-sk trees

TL;DR

This paper constructs a combinatorial bijection on di-sk trees, revealing new distributional symmetries of permutation statistics and providing alternative proofs for known equidistribution results in pattern-avoiding permutations.

Contribution

It introduces a novel bijection on di-sk trees that demonstrates distributional equivalences of permutation statistics, offering new insights and alternative proofs for classical results.

Findings

The bijection proves the equidistribution of certain permutation statistic quintuples.

Specialization to 312-avoiding permutations yields an alternative proof of a known equidistribution result.

Additional results on tree traversal statistics are presented.

Abstract

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. Rubey's 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.