The maximum multiplicity of a generator in a reduced word

We study the maximum multiplicity M(k, n) of a simple transposition s_k = (k k+1) in a reduced word for the longest permutation w₀ = n n-1… 2 1, a problem closely related to much previous work on sorting networks and on the "k-sets" problem. After reinterpreting the problem in terms of monotone weakly separated paths, we show that, for fixed k and growing n, the optimal collections are periodic in a precise sense, so that (Formula Presented) for a periodic function p_k and constant c_k. In fact we show that c_k is always rational, and compute several bounds and exact values for this quantity.