  In earlier work in collaboration with Pavel Galashin and Thomas McConville we
introduced a version of chip-firing for root systems. Our investigation of root
system chip-firing led us to define certain polynomials analogous to Ehrhart
polynomials of lattice polytopes, which we termed the symmetric and truncated
Ehrhart-like polynomials. We conjectured that these polynomials have
nonnegative integer coefficients. Here we affirm "half" of this positivity
conjecture by providing a positive, combinatorial formula for the coefficients
of the symmetric Ehrhart-like polynomials. This formula depends on a subtle
integrality property of slices of permutohedra, and in turn a lemma concerning
dilations of projections of root polytopes, which both may be of independent
interest. We also discuss how our formula very naturally suggests a conjecture
for the coefficients of the truncated Ehrhart-like polynomials that turns out
to be false in general, but which may hold in some cases.
