  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
$\det_{0\le i,j\le n-1}\big(\om \delta_{ij}+\binom {m+i+j}j\big)$, 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 $\om=1$.
