It seems like from here that chains in the brahut in $S_n$ order are length $\binom{n}{2}$ and there is a bijection with young tableaux of shape $\lambda = (n-1, n-2, \ldots,1)$.
Knuth in problem 38 seems to say there is a bijection between young tableaux of shape $\lambda$ and primitive sorting networks (where all of the comparators are ordered in a list).
It seems like the partial product of the comparitors (when applied to $t$ the order reversing permutation as transpositions), produces a chain in the brahut ordering.
${\rm inv}(t) \supset {\rm inv}(\tau_1 t) \supset{\rm inv}(\tau_2 \tau_1 t) \supset ... \supset {\rm inv}(e) = \emptyset$
Where
- $t \in S_n$ sends $t(k) = n-k+1$.
- $e$ is the identity element of $S_n$,
- $\tau_i$ is the transposition associated the $i$th comparator.
- ${\rm inv}(\sigma) = \{(i, j) : 1 \leq i < j \leq n {\rm and } \sigma(i) > \sigma(j)\}$ for $\sigma \in S_n$
Is there a tidy way to state this bijection with young tableaux of shape $\lambda$?