  We consider the triangular lattice ice model (20-Vertex model) with four
types of domain-wall type boundary conditions. In types 1 and 2, the
configurations are shown to be equinumerous to the quarter-turn symmetric
domino tilings of an Aztec-like holey square, with a central cross-shaped hole.
The proof of this statement makes extensive use of integrability and of a
connection to the 6-Vertex model. The type 3 configurations are conjectured to
be in same number as domino tilings of a particular triangle. The four
enumeration problems are reformulated in terms of four types of Alternating
Phase Matrices with entries 0 and sixth roots of unity, subject to suitable
alternation conditions. Our result is a generalization of the ASM-DPP
correspondence. Several refined versions of the above correspondences are also
discussed.
