Shelling the \(m=1\) amplituhedron

The amplituhedron \(\mathcal{A}_{n,k,m}\) was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes planar \(\mathcal{N}=4\) supersymmetric Yang-Mills theory. It is projection inside the Grassmannian \(\text{Gr}_{k,k+m}\) of totally nonnegative part \(\text{Gr}_{k,n}\). Karp Williams (2019) studied \(m=1\) \(\mathcal{A}_{n,k,1}\), giving regular CW decomposition it. Its face poset \(R_{n,l}\) (with \(l := n-k-1\)) consists all projective sign vectors length \(n\) with exactly \(l\) changes. We show that EL-shellable, resolving problem posed Williams. This gives new proof \(\mathcal{A}_{n,k,1}\) homeomorphic closed ball, which originally proved also explicit formulas \(f\)-vector \(h\)-vector \(R_{n,l}\), it rank-log-concave strongly Sperner. Finally, we consider related \(P_{n,l}\) Machacek (2019), consisting at most rank-log-concave, conjecture Sperner.Mathematics Subject Classifications: 06A07, 14M15, 81T60, 05A19Keywords: Amplituhedron, shellability, Eulerian number, log concavity, Sperner property