  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 $Gr_{k,k+m}$ of the totally nonnegative part
of $Gr_{k,n}$. 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.
