Stack-sorting simplices: geometry and lattice-point enumeration

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 $2$.