Enumeration of lozenge tilings of hexagons with a central triangular hole

Mihai Ciucu, Theresia Eisenkölbl, Christian Krattenthaler and Douglas Zare


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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