Asymptotic lengths of permutahedra and associahedra

TL;DR

We address how non-shortness manifests in oriented polytopes by defining asymptotic $k$-lengths and asymptotic total length through face-chain excess. For permutahedra with the weak Bruhat orientation, we construct zebra chains and prove $\alpha_k=1$ for all $k$, giving $\alpha=1$. For associahedra with the Tamari orientation, thuja chains yield $\alpha_k=(k-2)/(2k-2)$ for $k>2$ and $\alpha=1/2$, showing a sharp contrast in asymptotic behavior. These results quantify how far these polytopes are from being short and illuminate the structure of higher coherences for non-coassociative diagonals.

Abstract

We define asymptotic lengths for families of oriented polytopes. We show that permutahedra with weak order orientations have asymptotic total length 1 and associahedra with Tamari order orientations have asymptotic total length 1/2.

Figures (7)

• Figure 1: The polytope $B^3(3)$
• Figure 2: The zebra chain $\mathbf{Zeb}^4(3,3)$
• Figure 3: The inductive step in the proof of Prop. \ref{['zebradim']}: green points join existing (gray) groups and red points form the $l-1$ new (blue) groups
• Figure 4: Partial zebra chain $\mathbf{F}^4(7)$
• Figure 5: A generating relation for Tamari order on the vertices of $K(6)$

Theorems & Definitions (22)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Proposition 1.4
• proof
• Definition 1.5
• Proposition 1.6
• proof
• Theorem 2.1
• Definition 2.2