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Permutohedron's volume via Dyck paths

Damian de la Fuente

TL;DR

This work provides a Dyck-path expression for the volume of the type $A_n$ permutohedron, connecting it to the sizes of lower Bruhat intervals in affine Weyl groups. It recasts the pyramid-volume formula into a polynomial recurrence $\mathcal{V}_n=\sum_{i=1}^n \Gamma'_{n,i} \mathcal{V}_{i-1} \mathcal{V}_{n-i}[i]$ and then interprets the resulting sum as over $n$-Dyck paths by defining $\Gamma'_D$ for each path, proving $\mathcal{V}_n=\sum_{D\in\mathcal{D}_n} \Gamma'_D$. This approach yields a short alternative proof of Postnikov's formula, clarifies the Catalan-number origin, and shows the volume is a polynomial in the dominant coordinates without Brion's formula. It also highlights a natural bijection between $n$-Dyck paths and rooted planar binary trees, linking permutohedron geometry with classical combinatorics.

Abstract

In a recent project, Castillo, Libedinsky, Plaza, and the author established a deep connection between the size of lower Bruhat intervals in affine Weyl groups and the volume of the permutohedron, showing that the former can be expressed as a linear combination of the latter. In this paper, we provide a formula for the volume of this polytope in terms of Dyck paths. Thus, we present a shorter, alternative, and enlightening proof of a previous formula given by Postnikov.

Permutohedron's volume via Dyck paths

TL;DR

This work provides a Dyck-path expression for the volume of the type permutohedron, connecting it to the sizes of lower Bruhat intervals in affine Weyl groups. It recasts the pyramid-volume formula into a polynomial recurrence and then interprets the resulting sum as over -Dyck paths by defining for each path, proving . This approach yields a short alternative proof of Postnikov's formula, clarifies the Catalan-number origin, and shows the volume is a polynomial in the dominant coordinates without Brion's formula. It also highlights a natural bijection between -Dyck paths and rooted planar binary trees, linking permutohedron geometry with classical combinatorics.

Abstract

In a recent project, Castillo, Libedinsky, Plaza, and the author established a deep connection between the size of lower Bruhat intervals in affine Weyl groups and the volume of the permutohedron, showing that the former can be expressed as a linear combination of the latter. In this paper, we provide a formula for the volume of this polytope in terms of Dyck paths. Thus, we present a shorter, alternative, and enlightening proof of a previous formula given by Postnikov.

Paper Structure

This paper contains 8 sections, 8 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The root system of type Lg
  4. Volumes and polytopes
  5. The permutohedron
  6. The result
  7. Step 1
  8. Step 2

Key Result

Lemma 2.3

Let $O\in\mathsf{P}\subset U$ be a convex polytope $\mathsf{P}$ with an interior point $O$, sitting in a $k$-dimensional subspace $U$ of $\mathbb{R}^{n+1}$. Then where $d(O,\mathsf{F})$ is the distance from $O$ to the space spanned by $\mathsf{F}$.

Figures (3)

  • Figure 1: The polynomial of a north step
  • Figure 2: The polynomial of a Dyck path
  • Figure 3: A $7$-Dyck path and a north step.

Theorems & Definitions (22)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 12 more