2023/08/31 by Eon Lee, Lee, Eon, Carson Mitchell +3
Mathematics · #05A15 #52B05 #52B20 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2308.16457
openalex publication_date 2023/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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⋯ 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⋯ n1. This then allows us to show that those simplices are Gorenstein of index 2.