Plane partitions and rowmotion on rectangular and trapezoidal posets

TL;DR

This paper introduces a birational map connecting labelings of rectangular and trapezoidal posets, providing a new bijective proof of plane partition enumeration and demonstrating the finite order of birational rowmotion on trapezoidal posets.

Contribution

It defines a novel birational map that tropicalizes to a bijection between plane partitions, resolving a conjecture about rowmotion's order on trapezoidal posets.

Findings

Bijection between plane partitions of rectangular and trapezoidal posets.

Proof that birational rowmotion on trapezoidal posets has finite order.

Resolution of a conjecture by Williams regarding rowmotion equivariance.

Abstract

We define a birational map between labelings of a rectangular poset and its associated trapezoidal poset. This map tropicalizes to a bijection between the plane partitions of these posets of fixed height, giving a new bijective proof of a result by Proctor. We also show that this map is equivariant with respect to birational rowmotion, resolving a conjecture of Williams and implying that birational rowmotion on trapezoidal posets has finite order.