Lozenge tilings of a halved hexagon with an array of triangles removed from the boundary, part II

TL;DR

This paper extends the enumeration of lozenge tilings of halved hexagons with boundary triangles removed, focusing on the staircase side, and provides explicit formulas for these tilings.

Contribution

It generalizes previous results by enumerating lozenge tilings with boundary triangles removed from the staircase side of a halved hexagon.

Findings

Derived explicit formulas for tilings with triangles removed from the staircase side.

Unified previous enumeration results into a broader generalization.

Provided formulas for hexagons with missing triangles on the symmetry axis.

Abstract

Proctor's work on staircase plane partitions yields an enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Rohatgi later extended this tiling enumeration to a halved hexagon with a triangle cut off from the boundary. In the previous paper, the author proved a common generalization of Proctor's and Rohatgi's results by enumerating lozenge tilings of a halved hexagon in the case an array of an arbitrary number of triangles has been removed from a non-staircase side. In this paper, we consider the other case when the array of triangles has been removed from the staircase side of the halved hexagon. Our result also implies an explicit formula for the number of tilings of a hexagon with an array of triangles missing on the symmetry axis.