On separable permutations and three other pairs in the Schr\"oder class

TL;DR

This paper analyzes positional statistics in four pattern-avoiding permutation classes related to Schr"oder numbers, deriving explicit generating functions and revealing connections to known combinatorial arrays.

Contribution

It introduces a unified approach using structural decompositions and the kernel method to derive explicit formulas for positional statistics in these classes, providing new insights and proofs.

Findings

Explicit multivariate generating functions for each class

Alternative proofs that classes are counted by Schr"oder numbers

Connections to binomial coefficients and Kreweras' sequences

Abstract

We study positional statistics for four families of pattern-avoiding permutations counted by the large Schr\"oder 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 Schr\"oder numbers. As a byproduct, we obtain alternative proofs that each of these classes is enumerated by the Schr\"oder 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.