This must be in the literature, but I wasn't able to find it with Google: For each positive integer n > 0, what is the maximum value of N for which there exists an n-by-n stochastic matrix A and a probability vector v such that A^N v is the first forward image of v under repeated multiplication by A that lies in the open positive orthant (i.e., has all its components positive)? This can be recast as a purely combinatorial question. Suppose a finite directed graph with n vertices has the property that for every pair of vertices v,w, there is a way to get from v to w in N moves. Define the "inconvenience" of the digraph as the smallest N that works for all v,w. How large can the inconvenience of an n-vertex directed graph be? (One must assume that the digraph has finite inconvenience to begin with.) I can prove that the answer lies somewhere between quadratic-in-n and exponential-in-n, and I suspect that the former is where the truth lies. But my particular guess for quadratic truth must be sub-optimal, because the associated integer sequence isn't in Sloane. Jim Propp