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Vertices of the monotone path polytopes of hypersimplicies

Germain Poullot

TL;DR

This work develops a detailed framework for monotone path polytopes, focusing on hypersimplices and generic directions. It establishes four explicit constructions of the monotone path polytope, then proves a necessary coherence criterion (enhanced steps) and, for the second hypersimplex, sufficiency via a novel lattice-path model and an induction scheme. The authors derive an exact vertex count for the monotone path polytope of $ ext{HypSimpl}[n][2]$ and classify coherent monotone paths by length, revealing exponential growth in coherent paths relative to total monotone paths and identifying a maximal coherent path length of $\left\lfloor\frac{3}{2}(n-1)\right\rfloor$. They also provide algorithmic and computational tools, discuss unimodality conjectures, and outline open questions for higher $k$ and non-generic directions, linking fiber polytopes, shadow vertex rules, and lattice-path combinatorics in a unified framework.

Abstract

The monotone path polytope of a polytope $P$ encapsulates the combinatorial behavior of the shadow vertex rule (a pivot rule used in linear programming) on $P$. Computing monotone path polytopes is the entry door to the larger subject of fiber polytopes, for which explicitly computing examples remains a challenge. We first give a detailed presentation on how to construct monotone path polytopes. Monotone path polytopes of cubes and simplices have been known since the seminal article of Billera and Sturmfels. We extend these results to hypersimplices by linking this problem to the combinatorics of lattice paths. Indeed, we give a combinatorial model which describes the vertices of the monotone path polytope of the hypersimplex $Δ(n, 2)$ (for any generic direction). With this model, we give a precise count of these vertices, and furthermore count the number of coherent monotone paths on $Δ(n, 2)$ according to their lengths. We prove that some of the results obtained also hold for hypersimplices $Δ(n, k)$ for $k\geq 2$.

Vertices of the monotone path polytopes of hypersimplicies

TL;DR

This work develops a detailed framework for monotone path polytopes, focusing on hypersimplices and generic directions. It establishes four explicit constructions of the monotone path polytope, then proves a necessary coherence criterion (enhanced steps) and, for the second hypersimplex, sufficiency via a novel lattice-path model and an induction scheme. The authors derive an exact vertex count for the monotone path polytope of and classify coherent monotone paths by length, revealing exponential growth in coherent paths relative to total monotone paths and identifying a maximal coherent path length of . They also provide algorithmic and computational tools, discuss unimodality conjectures, and outline open questions for higher and non-generic directions, linking fiber polytopes, shadow vertex rules, and lattice-path combinatorics in a unified framework.

Abstract

The monotone path polytope of a polytope encapsulates the combinatorial behavior of the shadow vertex rule (a pivot rule used in linear programming) on . Computing monotone path polytopes is the entry door to the larger subject of fiber polytopes, for which explicitly computing examples remains a challenge. We first give a detailed presentation on how to construct monotone path polytopes. Monotone path polytopes of cubes and simplices have been known since the seminal article of Billera and Sturmfels. We extend these results to hypersimplices by linking this problem to the combinatorics of lattice paths. Indeed, we give a combinatorial model which describes the vertices of the monotone path polytope of the hypersimplex (for any generic direction). With this model, we give a precise count of these vertices, and furthermore count the number of coherent monotone paths on according to their lengths. We prove that some of the results obtained also hold for hypersimplices for .

Paper Structure

This paper contains 23 sections, 19 theorems, 20 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on monotone paths polytopes
  3. Explicit construction(s) of monotone path polytopes
  4. Exploring the space of ${\boldsymbol{\omega}}$, and projection of the normal fan of $\mathsf{P}$
  5. Line bundle and stereographic projection
  6. Convex hull of (explicit) points
  7. Minkowski sum of sections
  8. Coherent paths on hypersimplices
  9. A necessary criterion for coherent paths on $\HypSimpl$
  10. Enhanced steps criterion
  11. Comparison with coherent paths for a non-generic direction
  12. Sufficiency of the enhanced steps criterion in the case $\HypSimpl[n][2]$
  13. From coherent paths to lattice paths
  14. The induction process
  15. Counting the number of coherent monotone paths on $\HypSimpl[n][2]$
  16. ...and 8 more sections

Key Result

Theorem 2.4

(BilleraSturmfels-FiberPolytope). The face lattice of $\MPP$ is the lattice of coherent cellular strings on $\mathsf{P}$.

Figures (12)

  • Figure 1: Animated construction of the normal fan of the monotone path polytope of the 3-dimensional simplex. For each ${\boldsymbol{\omega}}\in \mathbb{R}^3$ orthogonal to ${\boldsymbol{c}}$, we project $\mathsf{\Delta}_3$ onto the plane spanned by $({\boldsymbol{c}},{\boldsymbol{\omega}})$ (Right), and record the corresponding coherent monotone path (Left). (Animated figures obviously do not display on paper, please use a PDF viewer (like Adobe Acrobat Reader), or go on my personal website, or ask by email.)
  • Figure 2: (Top) For reference, the tetrahedron $\mathsf{P} = \mathsf{\Delta}_3$, and the direction ${\boldsymbol{c}}$ from \ref{['fig:AnimatedMPPSimplex']}. (Left) The stereographic projection of the normal fan of $\mathsf{P}$, each colored region correspond to (the normal cone of) a vertex of $\mathsf{P}$. (Right) Two rays give rise to the same coherent monotone path if and only they intersect the same colored regions, we draw the resulting fan, labeled accordingly.
  • Figure 3: The construction of $\MPP$ as a sum of sections for the tetrahedron $\mathsf{P} = \mathsf{\Delta}_3$. Each section is orthogonal to ${\boldsymbol{c}}$ and contains a vertex (except for ${\boldsymbol{v}}_{\min}$ and ${\boldsymbol{v}}_{\max}$).
  • Figure 4: (Left) The $(4,2)$-hypersimplex lives in the hyperplane $\{{\boldsymbol{x}} ~;~ \sum_{i=1}^4 x_i = 2\}$ inside $\mathbb{R}^4$. (Middle) The monotone path polytope $\MPPHypSimpl[4][2]$ is an octagon, each vertex of which is labeled by the corresponding (coherent) monotone path, drawn on $\HypSimpl[4][2]$. (Right) Actually, $\mathsf{M}(4,2)$ is not a regular octagon, but the octagon depicted here, the two crosses correspond to the two monotone paths on $\HypSimpl[4][2]$ which are not coherent.
  • Figure 5: (Left) For the given ${\boldsymbol{c}}$ and ${\boldsymbol{\omega}}$, the hypersimplex $\HypSimpl[5][2]$ is projected onto the 10 points drawn, where each vertex of $\HypSimpl[5][2]$ is indicated by its support. The coherent path $P$ captured is drawn in blue. (Middle and Right) $P$ corresponds to the diagonal-avoiding path depicted on the right, while associating $P$ to lattice points $(x,y)$ with $x < y$ give the middle figure.
  • ...and 7 more figures

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5: Folklore
  • Remark 2.6
  • proof : Proof of \ref{['thm:MPPSumSectionAtVertices']}
  • Theorem 2.7: BilleraSturmfels-FiberPolytope
  • Theorem 2.8: BilleraSturmfels-FiberPolytope
  • Definition 2.9
  • ...and 44 more