# A new bijective proof of Babson and Steingr\'{\i}msson's conjecture

## TL;DR

This paper presents a novel bijective proof confirming the equidistribution of two permutation statistics, $stat$ and $maj$, resolving a conjecture by Babson and Steingrímsson with a combinatorial approach.

## Contribution

The paper introduces a new bijective proof of Babson and Steingrímsson's conjecture, providing a combinatorial alternative to previous algebraic and computational proofs.

## Key findings

- Confirmed that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$.
- Provided a new combinatorial bijection proof of the conjecture.
- Enhanced understanding of permutation statistics and their distributions.

## Abstract

Babson and Steingr\'{\i}msson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Given a permutation $\pi$, let $des(\pi)$ denote the descent number of $\pi$ and $maj(\pi)$ denote the major index of $\pi$. Babson and Steingr\'{\i}msson conjectured that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$. Foata and Zeilberger settled this conjecture using q-enumeration, generating functions and Maple packages ROTA and PERCY. Later, Burstein provided a bijective proof of a refinement of this conjecture. In this paper, we give a new bijective proof of this conjecture.