  We introduce and study the domain wall boundary partition function of the
integrable six-vertex model with triangular boundary. We first formulate the
domain wall boundary partition function with triangular boundary by using the
$U_q(sl_2)$ $R$-matrix and a special class of the triangular $K$-matrix. By
using its graphical representation, we make the Izergin-Korepin analysis with
the help of the Yang-Baxter relation and the reflection equation to give a
characterization of the partition function. The explict form of the symmetric
function representing the partition function is presented by showing that it
satisfies all the required properties.
