Involutions under Bruhat order and labeled Motzkin Paths

TL;DR

This paper introduces a new statistic on Motzkin paths that captures the rank generating function of involutions under Bruhat order, using bijections and interpretations involving visible inversions, with applications to fixed-point-free involutions and connections to the Ethiopian dinner game.

Contribution

It presents a novel statistic on Motzkin paths linked to Bruhat order involutions, providing new proofs and insights into their combinatorial structure.

Findings

Established a bijection between permutations and labeled Motzkin paths.

Derived a new expression for the rank generating function of involutions.

Connected the combinatorics of involutions to the Ethiopian dinner game.

Abstract

In this note, we introduce a statistic on Motzkin paths that describes the rank generating function of Bruhat order for involutions. Our proof relies on a bijection introduced by Philippe Biane from permutations to certain labeled Motzkin paths and a recently introduced interpretation of this rank generating function in terms of visible inversions. By restricting our identity to fixed-point-free (FPF) involutions, we recover an identity due to Louis Billera, Lionel Levine and Karola M\'esz\'aros with a previous bijective proof by Matthew Watson. Our work sheds new light on the Ethiopian dinner game.