Bijective recurrences concerning two Schröder triangles
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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=0^n-12^jr(n-1,j), r(n,k) =r(n-1,k-1)+∑_j=k^n-12^j-kr(n-1,j), 1< k< n, s(n,0) =∑_j=1^n-12·3^j-1s(n-1,j), s(n,k) =s(n-1,k-1)+∑_j=k+1^n-12·3^j-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.
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