Combinatorial nature of ground state vector of O(1) loop model

TL;DR

This paper presents numerical evidence linking the ground state vector of the dense O(1) loop model to the counts of states in the fully packed loop model with fixed link-patterns, suggesting a deep combinatorial connection.

Contribution

It conjectures a new relationship between the ground state components of the O(1) loop model and combinatorial counts of fully packed loop states, extending previous conjectures.

Findings

Numerical evidence supports the conjectured correspondence.

The conjecture generalizes previous related conjectures.

Provides a new perspective on the combinatorial structure of the ground state vector.

Abstract

Hanging about a hypothetical connections between the ground state vector for some special spin systems and the alternating-sign matrices, we have found a numerical evidence for the fact that the numbers of the states of the fully packed loop model with fixed link-patterns coincide with the components of the ground state vector of the dense O$(1)$ loop model considered by Batchelor, de Gier and Nienhuis. Our conjecture generalizes in a sense the conjecture of Bosley and Fidkowski, refined by Cohn and Propp, and proved by Wieland.