Enumeration of lozenge tilings of hexagons with a central triangular hole

Abstract

We deal with unweighted and weighted enumerations of lozenge tilings of a hexagon with side lengths $a,b+m,c,a+m,b,c+m$, where an equilateral triangle of side length $m$ has been removed from the center. We give closed formulas for the plain enumeration and for a certain $(-1)$-enumeration of these lozenge tilings. In the case that $a=b=c$, we also provide closed formulas for certain weighted enumerations of those lozenge tilings that are cyclically symmetric. For $m=0$, the latter formulas specialize to statements about weighted enumerations of cyclically symmetric plane partitions. One such specialization gives a proof of a conjecture of Stembridge on a certain weighted count of cyclically symmetric plane partitions. The tools employed in our proofs are nonstandard applications of the theory of nonintersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants $\det_{0\le i,j\le n-1}\big(\om δ_{ij}+\binom {m+i+j}j\big)$, where $\om$ is any 6th root of unity. These determinant evaluations are variations of a famous result due to Andrews (Invent. Math. 53 (1979), 193--225), which corresponds to $\om=1$.