  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 in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. It is a
projection inside the Grassmannian $\text{Gr}_{k,k+m}$ of the totally
nonnegative part of $\text{Gr}_{k,n}$. Karp and Williams (2019) studied the
$m=1$ amplituhedron $\mathcal{A}_{n,k,1}$, giving a regular CW decomposition of
it. Its face poset $R_{n,l}$ (with $l := n-k-1$) consists of all projective
sign vectors of length $n$ with exactly $l$ sign changes. We show that
$R_{n,l}$ is EL-shellable, resolving a problem posed by Karp and Williams. This
gives a new proof that $\mathcal{A}_{n,k,1}$ is homeomorphic to a closed ball,
which was originally proved by Karp and Williams. We also give explicit
formulas for the $f$-vector and $h$-vector of $R_{n,l}$, and show that it is
rank-log-concave and strongly Sperner. Finally, we consider a related poset
$P_{n,l}$ introduced by Machacek (2019), consisting of all projective sign
vectors of length $n$ with at most $l$ sign changes. We show that it is
rank-log-concave, and conjecture that it is Sperner.
