Gamma-positivity for octopuses: a bijective proof

Abstract

Chapoton introduced an interesting family of polytopes called arbor polytopes, where a polytope $\mathcal{Q}_\tau$ is associated to any arbor $\tau$. Chapoton conjectured that the polynomial $h(\tau)$ which counts lattice points in $\mathcal{Q}_\tau$ by number of nonzero entries is palindromic and unimodal. Athanasiadis, Xiao, and Yan recently proved that $h(\tau)$ is gamma-positive for a certain class of arbors they call octopuses. Since their proof was computational, they asked for one that is bijective. We present such a proof. We also extend their results by showing that $h(\tau)$ is gamma-positive for a larger class of arbors we call lopsided octopuses.