  It is widely known that there is no sign problem in Path Integral Monte Carlo
(PIMC) simulations of fermions in one dimension. Yet, as far as the author is
aware, there is no direct proof of this in the literature. This work shows that
the $sign$ of the $N$-fermion anti-symmetric free propagator is given by the
product of all possible pairs of particle separations, or relative
displacements. For a non-vanishing closed-loop product of such propagators, as
required by PIMC, all relative displacements from adjacent propagators are
paired into perfect squares, and therefore the loop product must be positive,
but only in one dimension. By comparison, permutation sampling, which does not
evaluate the determinant of the anti-symmetric propagator exactly, remains
plagued by a low-level sign problem, even in one dimension.
