What is the longest element of $S_n$ as a product of adjacent transpositions?

Question

I can't seem to get this to work. According to wikipeda, the longest element of $S_n$ should be expressible as a product of $n(n-1)/2$ adjacent transpositions by $$ (n, n-1)(n-1,n-2)\cdots(21)(n-1,n-2)(n-2,n-3)\cdots $$ but I don't understand what pattern they're trying to imply. This should be the order reversing permutation, but even for $n=4$, I think I'm doing it wrong.

I think that pattern says, for $n=4$, the longest element should be expressible as $6$ adjacent transpositions $$ (43)(32)(21)(32)(21)(21)=(43)(32)(21)(32) $$ but that's not right since it fixes $2$ instead of sending $2$ to $3$. What am I reading wrong?

Answer 1

It should be

$$(n,n-1)(n-1,n-2)\cdots (2,1)(n,n-1)(n-1,n-2)\cdots(3,2) \cdots (n,n-1)(n-1,n-2)(n,n-1)$$

so for $S_4$ it is $(4,3)(3,2)(2,1)(4,3)(3,2)(4,3)$.

Answer 2

(Hint more than full answer)

Have a look at what is called the "permutohedron"

https://en.wikipedia.org/wiki/Permutohedron

whose vertices are permutations, edges are transpositions changing one permutation into another, with many geometrical properties reflecting group properties of $S_n$.

Your issue is likely to be mapped into a permutohedron issue; it looks like finding the longest sequence of edges that you can "assemble" as a connecting path between the identity permutation and the target permutation with the constraint of using edges that correspond to adjacent transpositions.