Resolving Two Conjectures on Staircase Encodings and Boundary Grids of $132$ and $123$-avoiding permutations

TL;DR

This paper proves two conjectures relating pattern avoidance in permutations to properties of associated graphs on staircase and boundary grids, advancing understanding of permutation classes and their graphical representations.

Contribution

It establishes the enumeration of a family of staircase encodings and characterizes when the downcore graph is pure based on pattern avoidance.

Findings

Enumeration of staircase encodings for certain permutation classes

Characterization of when the downcore graph is pure

Proof of two conjectures on pattern avoidance and graph properties

Abstract

This paper analyzes relations between pattern avoidance of certain permutations and graphs on staircase grids and boundary grids, and proves two conjectures posed by Bean, Tannock, and Ulfarsson (2015). More specifically, this paper enumerates a certain family of staircase encodings and proves that the downcore graph, a certain graph established on the boundary grid, is pure if and only if the permutation corresponding to the boundary grid avoids the classical patterns 123 and 2143.