Enumeration of lozenge tilings of hexagons with a central
triangular hole
Abstract
We deal with the unweighted and weighted
enumeration 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 forms
for the (plain) enumeration, and for a certain (-1)-enumeration of
these lozenge tilings. In the case that a=b=c,
we provide as well closed forms for certain
weighted enumerations of such lozenge tilings that are
cyclically symmetric. In the special case a=b=c, m=0,
we obtain
results about weighted enumerations of cyclically symmetric plane partitions.
Our tools in the proofs are (nonstandard)
applications of nonintersecting lattice paths, and determinant
evaluations. In particular, we evaluate the determinants
\det(\omega \delta_{ij}+\binom
{m+i+j}j), where \omega is any
6th root of unity. These determinant evaluations are variations of
a famous evaluation due to Andrews
(Invent. Math. 53 (1979), 193-225),
which is the case \omega=1.(57 pages)
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