On the Counting of Fully Packed Loop Configurations. Some new conjectures

TL;DR

This paper proposes new conjectures about the enumeration of Fully Packed Loop (FPL) configurations related to specific link patterns, utilizing the Razumov-Stroganov Ansatz and analyzing the Temperley-Lieb chain ground state for sizes up to 22.

Contribution

It introduces novel conjectures on FPL configuration counts based on ground state analysis, extending understanding of link patterns and boundary conditions.

Findings

Proposes new conjectures on FPL configuration numbers.

Analyzes ground states of the Temperley-Lieb chain up to size 22.

Utilizes the Razumov-Stroganov Ansatz for conjecture formulation.

Abstract

New conjectures are proposed on the numbers of FPL configurations pertaining to certain types of link patterns. Making use of the Razumov and Stroganov Ansatz, these conjectures are based on the analysis of the ground state of the Temperley-Lieb chain, for periodic boundary conditions and so-called ``identified connectivities'', up to size $2n=22$.