Bijective proofs of recurrences involving two Schröder triangles

Let r(n,k) (resp. s(n,k)) be the number of Schröder paths (resp. little Schröder paths) of length 2n with k hills, and set r(0,0)=s(0,0)=1. We bijectively establish the following recurrence relations: r(n,0)=∑j=0n−12jr(n−1,j),n≥1,r(n,k)=r(n−1,k−1)+∑j=kn−12j−kr(n−1,j),1≤k≤n,s(n,0)=∑j=1n−12⋅3j−1s(n−1,j),n≥1,s(n,k)=s(n−1,k−1)+∑j=k+1n−12⋅3j−k−1s(n−1,j),1≤k≤n. The infinite lower triangular matrices [r(n,k)]n,k≥0 and [s(n,k)]n,k≥0, whose row sums produce the large and little Schröder 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 Schröder 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≥0 as well.