A finite sequence of distinct real numbers of length $n$ determines a linear order of $\{1,\ldots,n\}$, by mapping position to rank; call this the *permutation of the sequence*.

Consider the following digraph $P_n$. The vertex set of $P_n$ comprises all the linear orders of $\{1,\ldots,n\}$. Then an arrow connects vertex $v$ to vertex $w$ exactly when the permutation of $v$'s terminal length $n-1$ subsequence equals the permutation $w$'s initial length $n-1$ subsequence.

**Question 1**: Do these digraphs occur in the literature, and perhaps have a standard terminology and notation?

**Observation**: I have verified the following (which surprised me) by exhaustive computation: $P_4$ admits 13063680 Hamiltonian paths originating from $v=(1,2,3,4)$ and every one of these extends to a Hamiltonian cycle!

**Question 2**: Is the previous observation a direct consequence of known theorem?

**Question 3**: Is anything known about the asymptotics of enumerations of Hamiltonian cycles in these digraphs?