  We study the open version of the su$(m|n)$ supersymmetric Haldane-Shastry
spin chain associated to the $BC_N$ extended root system. We first evaluate the
model's partition function by modding out the dynamical degrees of freedom of
the su$(m|n)$ supersymmetric spin Sutherland model of $BC_N$ type, whose
spectrum we fully determine. We then construct a generalized partition function
depending polynomially on two sets of variables, which yields the standard one
when evaluated at a suitable point. We show that this generalized partition
function can be written in terms of two variants of the classical skew super
Schur polynomials, which admit a combinatorial definition in terms of a new
type of skew Young tableaux and border strips (or, equivalently, extended
motifs). In this way we derive a remarkable description of the spectrum in
terms of this new class of extended motifs, reminiscent of the analogous one
for the closed Haldane-Shastry chain. We provide several concretes examples of
this description, and in particular study in detail the su$(1|1)$ model finding
an analytic expression for its Helmholtz free energy in the thermodynamic
limit.
