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On separable permutations and three other pairs in the Schröder class

Juan B. Gil, Oscar A. Lopez, Michael D. Weiner

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.

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.

Paper Structure

This paper contains 6 sections, 15 theorems, 52 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Separable permutations
  3. (1324,1423)-avoiding permutations
  4. (1423,2413)-avoiding permutations
  5. (1324,2134)-avoiding permutations
  6. Positional statistics for $1\prec n$

Key Result

Proposition 2.1

The function $g(x,u)=\sum\limits_{n=1}^\infty\sum\limits_{\ell=1}^n\, \lvert\mathcal{S}_n^{\ell\mapsto 1}(2413,3142)\rvert\, u^{\ell} x^n$ satisfies A few terms of $\lvert\mathcal{S}_n^{\ell\mapsto 1}(2413,3142)\rvert$ are listed in Table tab:byPos1.

Figures (1)

  • Figure 1: Graphical representation of $\phi_3$.

Theorems & Definitions (31)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 3.1
  • proof
  • Remark
  • ...and 21 more