Forced perimeter in Elnitksy polygons

TL;DR

This paper investigates the conditions under which certain perimeter tiles are forced in all rhombic tilings of Elnitsky polygons, linking forcing to 321-patterns and enumerating permutations with many forced tiles using Catalan numbers.

Contribution

It characterizes when perimeter tiles are forced in all tilings, relates forcing to 321-patterns, and enumerates permutations with maximal forced tiles using Catalan numbers.

Findings

Forced perimeter tiles are characterized by 321-patterns.

Permutations with the most forced perimeter tiles are counted by Catalan numbers.

The study provides a complete characterization of forcing in Elnitsky polygons.

Abstract

We study tiling-based perimeter and characterize when a given perimeter tile appears in all rhombic tilings of an Elnitsky polygon. Regardless of where on the perimeter this tile appears, its forcing can be described in terms of 321-patterns. We characterize the permutations with maximally many forced right-perimeter tiles, and show that they are enumerated by the Catalan numbers.