# Bijective proofs of shuffle compatibility results

## TL;DR

This paper introduces bijective methods to prove shuffle compatibility of permutation statistics, simplifying previous algebraic proofs and enabling the construction of new proofs, including resolving a conjecture by Gessel and Zhuang.

## Contribution

It presents a bijective approach to demonstrate shuffle compatibility, offering a systematic method for proofs and extending understanding of permutation statistics.

## Key findings

- Many permutation statistics are shuffle compatible via simple bijections
- A new method for constructing bijective proofs is developed
- Confirmed Gessel and Zhuang's conjecture on shuffle compatibility

## Abstract

Define a permutation to be any sequence of distinct positive integers. Given two permutations p and s on disjoint underlying sets, we denote by p sh s the set of shuffles of p and s (the set of all permutations obtained by interleaving the two permutations). A permutation statistic is a function St whose domain is the set of permutations such that St(p) only depends on the relative order of the elements of p. A permutation statistic is shuffle compatible if the distribution of St on p sh s depends only on St(p) and St(s) and their lengths rather than on the individual permutations themselves. This notion is implicit in the work of Stanley in his theory of P-partitions. The definition was explicitly given by Gessel and Zhuang who proved that various permutation statistics were shuffle compatible using mainly algebraic means. This work was continued by Grinberg. The purpose of the present article is to use bijective techniques to give demonstrations of shuffle compatibility. In particular, we show how a large number of permutation statistics can be shown to be shuffle compatible using a few simple bijections. Our approach also leads to a method for constructing such bijective proofs rather than having to treat each one in an ad hoc manner. Finally, we are able to prove a conjecture of Gessel and Zhuang about the shuffle compatibility of a certain statistic.