  Say that a permutation of $1,2,\ldots,n$ is \textit{$k$-bounded} if every
pair of consecutive entries in the permutation differs by no more than $k$.
Such a permutation is \textit{anchored} if the first entry is $1$ and the last
entry is $n$. We show that the generating function for the enumeration of
$k$-bounded anchored permutations is always rational, mirroring the known
result on (non-anchored) $k$-bounded permutations due to Avgustinovich and
Kitaev. We then explicitly determine the recursive formulas of minimal depth
for the number of anchored $k$-bounded permutations of $n$ for $k=2$ and $k=3$,
resolving a conjecture listed on the Online Encyclopedia of Integer Sequences
(entry A249665).
  We additionally show that the number of anchored $k$-bounded permutations of
$n$ is asymptotically $O\left(k^n\right)$ as a function of $n$ for a given $k$.
