Dimensions of toggleability spaces

TL;DR

This paper proves a conjecture about the dimensions of toggleability spaces for certain posets, showing they equal the rank plus one for a broad class of posets, using rook statistics techniques.

Contribution

It generalizes the dimension results of toggleability spaces to a wider family of posets defined by restricted diagrams, extending previous specific cases.

Findings

Confirmed the conjecture for product of chains, shifted staircases, and type-A and B root posets.

Established that toggleability space dimensions equal poset rank plus one for new classes of posets.

Utilized rook statistics techniques to achieve these results.

Abstract

We establish a conjecture of Defant, Hopkins, Poznanovi\'{c}, and Propp concerning the dimensions of toggleability spaces for products of chains, shifted staircases, type-A root posets, and type-B posets. Generalizing this result, we show that for a larger family of posets defined by restricted diagrams, the dimensions of toggleability spaces are equal to the rank of the poset plus one. As part of our approach, we build upon the technique of rook statistics introduced by Chan, Haddadan, Hopkins, and Moci.