Stack-sorting simplices: geometry and lattice-point enumeration

TL;DR

This work studies subpolytopes of the permutahedron that arise as convex hulls of permutations generated by iterations of the stack-sorting algorithm, focusing on Ln1 permutations and a distinguished $\tau_n=23\cdots n1$. The authors prove that the stack-sorting polytopes conv$(\mathcal{S}^{\boldsymbol{\pi}})$ are $(n-1)$-simplices for $\boldsymbol{\pi}\in\mathcal{L}^n$, and they analyze the specific simplex $\triangle_n=\operatorname{conv}(\mathcal{S}^{\boldsymbol{\tau}_n})$, showing it is hollow and integrally equivalent to the $(n-1)$-dimensional lecture-hall simplex $P_{n-1}$. They derive the Ehrhart theory for $\triangle_n$, obtaining $L_{\mathbb{Z}}(\triangle_n;t)=(t+1)^{n-1}$ and an Eulerian-number-based Ehrhart series, and they establish that $\triangle_n$ is Gorenstein of index $2$. Additionally, the paper develops a real-lattice point framework with a recurrence linking higher- and lower-dimensional counts, providing a robust approach to understanding the lattice structure of these stack-sorting simplices and their geometric properties.

Abstract

We initiate the study of subpolytopes of the permutahedron that arise as the convex hulls of stack-sorting on permutations. We primarily focus on $Ln1$ permutations, i.e., permutations of length $n$ whose penultimate and last entries are $n$ and $1$, respectively. First, we present some enumerative results on $Ln1$ permutations. Then we show that the polytopes that arise from stack-sorting on $Ln1$ permutations are simplices and proceed to study their geometry and lattice-point enumeration. In addition, we pose questions and problems for further investigation. Particular focus is then taken on the $Ln1$ permutation $23\cdots n1$. We show that the convex hull of all its iterations through the stack-sorting algorithm shares the same lattice-point enumerator as that of the $(n-1)$-dimensional unit cube and lecture-hall simplex. Lastly, we detail some results on the real lattice-point enumerator for variations of the simplices arising from stack-sorting on the permutation $23\cdots n1$. This then allows us to show that those simplices are Gorenstein of index $2$.

Figures (7)

• Figure 1: A visualization of step $2$ of the stack-sorting algorithm as presented in Example \ref{['ex:ssa']}.
• Figure 2: Left to Right: $\Pi_2\subset\mathbb{R}^2$, $\Pi_3\subset\mathbb{R}^3$, and $\Pi_4\subset\mathbb{R}^4$.
• Figure 3: The first three integral dilates of the two-dimensional unit cube $\square^2$.
• Figure 4: The simplex $\mathop{\mathrm{conv}}\nolimits(\mathcal{S}^{321})$ in the $3$-permutahedron $\Pi_3$.
• Figure 5: From Left to Right: the $\mathcal{S}^{\boldsymbol{\tau}_4}$-simplex $\triangle_4\subseteq \mathbb{R}^4$, the 3-unit cube $\square^3$, and the 3-dimensional lecture-hall simplex $P_{3}$.

Theorems & Definitions (51)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Example 2.4
• Definition 2.5
• Proposition 2.6
• Definition 2.7
• Definition 2.8
• Example 2.9
• Proposition 2.10: Theorems 1 and 2.2,BrightSavage