  We study the maximum multiplicity $\mathcal{M}(k,n)$ of a simple
transposition $s_k=(k \: k+1)$ in a reduced word for the longest permutation
$w_0=n \: n-1 \: \cdots \: 2 \: 1$, a problem closely related to much previous
work on sorting networks and on the "$k$-set" problem. After reinterpreting the
problem in terms of monotone weakly separated paths, we show that, for fixed
$k$ and sufficiently large $n$, the optimal density is realized by paths which
are periodic in a precise sense, so that \[ \mathcal{M}(k,n)=c_k n + p_k(n) \]
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
with "repeatable patterns", which we introduce.
