  We introduce a framework to consider transport problems for integer-valued
random variables. We introduce weighting coefficients which allow us to
characterize transport problems in a gradient flow setting, and form the basis
of our introduction of a discrete version of the Benamou-Brenier formula.
Further, we use these coefficients to state a new form of weighted
log-concavity. These results are applied to prove the monotone case of the
Shepp-Olkin entropy concavity conjecture.
