  Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schr\"oder paths (resp. little
Schr\"oder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We
bijectively establish the following recurrence relations: \begin{align*}
r(n,0)&=\sum\limits_{j=0}^{n-1}2^{j}r(n-1,j),
r(n,k)&=r(n-1,k-1)+\sum\limits_{j=k}^{n-1}2^{j-k}r(n-1,j),\quad 1\le k\le n,
s(n,0) &=\sum\limits_{j=1}^{n-1}2\cdot3^{j-1}s(n-1,j), s(n,k)
&=s(n-1,k-1)+\sum\limits_{j=k+1}^{n-1}2\cdot3^{j-k-1}s(n-1,j),\quad 1\le k\le
n. \end{align*} The infinite lower triangular matrices $[r(n,k)]_{n,k\ge 0}$
and $[s(n,k)]_{n,k\ge 0}$, whose row sums produce the large and little
Schr\"oder numbers respectively, are two Riordan arrays of Bell type. Hence the
above recurrences can also be deduced from their $A$- and $Z$-sequences
characterizations.
  On the other hand, it is well-known that the large Schr\"oder numbers also
enumerate separable permutations. This propelled us to reveal the connection
with a lesser-known permutation statistic, called initial ascending run, whose
distribution on separable permutations is shown to be given by
$[r(n,k)]_{n,k\ge 0}$ as well.
