The
(tree) amplituhedron
\mathcal{A}_{n,k,m}
is the image in the Grassmannian
\mathrm {Gr}_{k,k+m}
of the totally nonnegative Grassmannian
\mathrm {Gr}_{k,n}^{\geq 0}
, under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in planar
\mathcal{N}=4
supersymmetric Yang–Mills theory. In the case relevant to physics (
m=4
), there is a collection of recursively-defined
4k
-dimensional
BCFW cells
in
\mathrm {Gr}_{k,n}^{\geq 0}
, whose images conjecturally "triangulate" the amplituhedron – that is, their images are disjoint and cover a dense subset of
\mathcal{A}_{n,k,4}
. In this paper, we approach this problem by first giving an explicit (as opposed to recursive) description of the BCFW cells. We then develop sign-variational tools which we use to prove that when
k=2
, the images of these cells are disjoint in
\mathcal{A}_{n,k,4}
. We also conjecture that for arbitrary even
m
, there is a decomposition of the amplituhedron
\mathcal{A}_{n,k,m}
involving precisely
M\big(k, n-k-m, \frac{m}{2}\big)
top-dimensional cells (of dimension
km
), where
M(a,b,c)
is the number of plane partitions contained in an
a \times b \times c
box. This agrees with the fact that when
m=4
, the number of BCFW cells is the Narayana number
N_{n-3,k+1} = \frac{1}{n-3}\binom{n-3}{k+1}\binom{n-3}{k}
.
