Enumeration of Lozenge Tilings of Hexagons with a Central Triangular Hole

We deal with the 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 nonin-tersecting lattice paths and determinant evaluations. In particular, we evaluate the determinants det 0i;jn?1 ? ! ij + ? m+i+j j , where ! is any 6th root of unity. These determinant evaluations are variations of a famous result due to Andrews (Invent.