On equidistribution theorem for plane partitions

TL;DR

This paper proves equidistribution results for statistics on plane partitions and introduces a new statistic, providing new visual proofs and insights into partition theory.

Contribution

It establishes equidistribution of key statistics on plane partitions and introduces the cohook area statistic, with novel visual proof techniques.

Findings

Proved equidistribution of (volume, trace) and (corner-hook volume, number of corners)

Introduced the cohook area statistic and proved its equidistribution with partition area

Provided a new visual proof for the volume generating function of plane partitions

Abstract

We prove equidistribution of two pairs of statistics on boxed plane partitions: (volume, trace) and (corner-hook volume, number of corners). The proof relies on different 3d visualizations of the corresponding non-intersecting path systems. In particular, we obtain a new visual proof for a volume generating function of plane partitions. We also introduce a new statistic called the cohook area on ordinary partitions, and prove that it is equidistributed with the area of partitions.