  The refined enumeration of alternating sign matrices (ASMs) of given order
having prescribed behavior near one or more of their boundary edges has been
the subject of extensive study, starting with the Refined Alternating Sign
Matrix Conjecture of Mills-Robbins-Rumsey, its proof by Zeilberger, and more
recent work on doubly-refined and triply-refined enumeration by several
authors. In this paper we extend the previously known results on this problem
by deriving explicit enumeration formulas for the "top-left-bottom"
(triply-refined) and "top-left-bottom-right" (quadruply-refined) enumerations.
The latter case solves the problem of computing the full boundary correlation
function for ASMs. The enumeration formulas are proved by deriving new
representations, which are of independent interest, for the partition function
of the square ice model with domain wall boundary conditions at the
"combinatorial point" 2{\pi}/3.
