Gamma vectors of partitioned permutohedra

TL;DR

The paper determines the γ-vectors of partitioned permutohedra $P_n(K)$, generalizing Foata–Schützenberger and linking to Athanasiadis' representation-theoretic description of the permutohedral cohomology. It proves a gamma-expansion $h_{P_n(K)}(t)=\sum_{j} \gamma_{n,j,K} t^j(1+t)^{n-1-2j}$ with des-counts on minimal coset reps, via a descent-preserving bijection between restricted cosets and parabolic cosets. The work unifies combinatorial and geometric viewpoints by showing the Foata–Schützenberger-type expansion specializes to the Eulerian polynomial when $K=\emptyset$ and corresponds to Athanasiadis' $\mathfrak{S}_n$-module decomposition of the permutohedral cohomology; it thereby verifies Gal’s conjecture for partitioned permutohedra. The results are grounded in the interplay of valley hopping, descent statistics, RS-K correspondence, and Kostka/Frobenius frameworks, creating a cohesive bridge between combinatorics and toric geometry.

Abstract

We determine that $γ$-vectors of partitioned permutohedra, thereby generalizing a result of Foata and Schützenberger. Our result is closely related to a result of Athanasiadis on the representation of the symmetric group on the cohomology of the permutohedral variety. We explain how to derive Athanasiadis' result from ours and vice versa.

Figures (7)

• Figure 1: Peaks, valleys and free letters for $w=672841359$.
• Figure 2: Valley-hopping.
• Figure 3: $w_j \in \widetilde{W^{K_j}}$ in Case (i).
• Figure 4: $w_j \in \widetilde{W^{K_j}}$ in Case (ii).
• Figure 5: The letter $k_j$ is a valley, namely the slope immediately after $k_j$ is an upslope.

Theorems & Definitions (29)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Proposition 2.1: HMSS, Proposition 7.4
• Theorem 3.1: FoataSchutzenberger
• Theorem 3.2
• Example 3.3
• Definition 4.1
• Example 4.2