Per Alexandersson: Snakes in a tableau

Let lambda, mu, and w be partitions. Let K(lambda, mu, w) be the number of semi-standard Young tableaux of skew shape lambda/mu and content w. We will give a combinatorial proof of the fact that K(n*lambda, n*mu, n*w) is non-zero iff K(lambda, mu, w) is non-zero, for all positive integers n.