  This paper analyzes relations between pattern avoidance of certain
permutations and graphs on staircase grids and boundary grids, and proves two
conjectures posed by Bean, Tannock, and Ulfarsson (2015). More specifically,
this paper enumerates a certain family of staircase encodings and proves that
the downcore graph, a certain graph established on the boundary grid, is pure
if and only if the permutation corresponding to the boundary grid avoids the
classical patterns 123 and 2143.
