Direct bijection between ASMs and 2-block diagonally symmetric lozenge tilings

Construct a direct bijection between alternating sign matrices of size n and r-block diagonally symmetric lozenge tilings of the hexagon H(2n,2n,n) in the special case r=(2,2,...,2), to explain the observed algebraic connection between the ASM enumeration and the count of these symmetric lozenge tilings.

Background

The paper introduces r-block diagonally symmetric lozenge tilings and derives product formulas for their weighted enumeration. In the special case r_i=2 for all i, the authors note that the resulting count is closely related to the number of alternating sign matrices (ASMs) of size n, via an expression involving Schur polynomials and Okada’s character formula.

Although the algebraic relationship explains the appearance of ASM numbers in this symmetric lozenge tiling context, the authors explicitly point out that a direct combinatorial correspondence between these tilings and ASMs has not yet been established.