Bijection between Increasing Binary Trees and Rook Placements on Double Staircases

TL;DR

This paper establishes a bijection between rook placements on double staircases and increasing binary trees, revealing connections to Eulerian polynomial statistics and extending previous bijections.

Contribution

It introduces new subclasses of rook placements and links their enumeration to Eulerian polynomial gamma-vectors, expanding the combinatorial understanding of these structures.

Findings

Enumeration of rook placements relates to gamma-vectors of Eulerian polynomials.

Established a bijection between rook placements and increasing binary trees.

Discussed generalizations and connections to existing bijections.

Abstract

In this paper, we shall construct a bijection between rook placements on double staircases (introduced by Josuat-Verg\`es in 2017) and increasing binary trees. We introduce two subclasses of rook placements on double staircases, which we call left and right-aligned rook placements. We show that their enumeration, while keeping track of a certain statistic, gives the $\gamma$-vectors of the Eulerian polynomials. We conclude with a discussion on a different bijection that fits in very well with our main bijection, and another discussion on generalising our main bijection. Our main bijection is a special case of a bijection due to Tewari (2019).