Words that give rise to an enumeration of elements of the symmetric group

Let $\mathbb{S}_m$ be the symmetric group on $m$ letters. Let $n=m-1$. Let $\mathbf{w}=a_1\cdots a_r$ be a word on the alphabet $\{1,\ldots,n\}$. We say that $\mathbf{w}$ gives rise to an enumeration of elements of the symmetric group $\mathbb{S}_m$ if $1,s_{a_r},s_{a_{r-1}}s_{a_r},\ldots,s_{a_2}\cdots s_{a_r},s_{a_1}\cdots s_{a_r}$ are all distinct elements of the symmetric group $\mathbb{S}_m$.

For example, $32323132323132323$ gives rise to an enumeration of elements of the symmetric group $\mathbb{S}_4$, and I think it is (one of) the longest possible such words.

Q. What is the length $r(n)$ of the longest word which gives rise to an enumeration of elements of the symmetric group $\mathbb{S}_m$. Is it always possible to find such a word of length $m!$, or only for $\mathbb{S}_3$?

Answer

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