Bijective recurrences concerning two Schröder triangles

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.