On the geometry of stack-sorting simplices

TL;DR

This paper proves that all stack-sorting polytopes are simplices, characterizes their volume and lattice points, and establishes bounds related to permutations, advancing understanding of their geometric structure.

Contribution

It demonstrates that all stack-sorting polytopes are simplices and provides volume and lattice point bounds for specific permutation-generated polytopes.

Findings

Stack-sorting polytopes are simplices.

Polytopes from Ln1 permutations have volume 1.

Polytopes from 2Ln1 permutations have no interior lattice points.

Abstract

We show that all stack-sorting polytopes are simplices. Furthermore, we show that the stack-sorting polytopes generated from $Ln1$ permutations have relative volume 1. We establish an upper bound for the number of lattice points in a stack-sorting polytope. In particular, stack-sorting polytopes generated from $2Ln1$ permutations have no interior points.