Doppelg\"angers: Bijections of Plane Partitions

TL;DR

This paper introduces a unified framework for bijective proofs showing certain posets, called doppelg"angers, have identical counts of P-partitions, and applies it to prove a classical plane partition enumeration theorem.

Contribution

It synthesizes algebraic and geometric techniques to establish bijections between posets, providing the first bijective proof of a known plane partition enumeration result.

Findings

Established a uniform framework for bijective proofs of doppelg"angers.

Provided the first bijective proof of Proctor's theorem on plane partitions.

Connected combinatorial bijections with geometric interpretations in flag varieties.

Abstract

We say two posets are "doppelg\"angers" if they have the same number of $P$-partitions of each height $k$. We give a uniform framework for bijective proofs that posets are doppelg\"angers by synthesizing $K$-theoretic Schubert calculus techniques of H. Thomas and A. Yong with M. Haiman's rectification bijection and an observation of R. Proctor. Geometrically, these bijections reflect the rational equivalence of certain subvarieties of minuscule flag manifolds. As a special case, we provide the first bijective proof of a 1983 theorem of R. Proctor---that plane partitions of height $k$ in a rectangle are equinumerous with plane partitions of height $k$ in a trapezoid.