On separable permutations and three other pairs in the Schröder class

Abstract: We study positional statistics for four families of pattern-avoiding permutations counted by the large Schröder numbers. Specifically, we focus on the pairs of patterns {2413,3142} (separable permutations), {1324,1423}, {1423,2413}, and {1324,2134}. For each class, we derive multivariate generating functions that track the relative positions of specific entries. Our approach combines structural decompositions with the kernel method to obtain explicit formulas involving the generating function for the Schröder numbers. As a byproduct, we obtain alternative proofs that each of these classes is enumerated by the Schröder numbers. We also identify several known triangular arrays arising from our positional refinements, including connections to the central binomial coefficients and sequences appearing in the work of Kreweras on covering hierarchies.