  We study properties of a non-Markovian random walk $X^{(n)}_l$, $l =0,1,2,
>...,n$, evolving in discrete time $l$ on a one-dimensional lattice of
integers, whose moves to the right or to the left are prescribed by the
\text{rise-and-descent} sequences characterizing random permutations $\pi$ of
$[n+1] = \{1,2,3, ...,n+1\}$. We determine exactly the probability of finding
the end-point $X_n = X^{(n)}_n$ of the trajectory of such a
permutation-generated random walk (PGRW) at site $X$, and show that in the
limit $n \to \infty$ it converges to a normal distribution with a smaller,
compared to the conventional P\'olya random walk, diffusion coefficient. We
formulate, as well, an auxiliary stochastic process whose distribution is
identic to the distribution of the intermediate points $X^{(n)}_l$, $l < n$,
which enables us to obtain the probability measure of different excursions and
to define the asymptotic distribution of the number of "turns" of the PGRW
trajectories.
