A q -enumeration of lozenge tilings of a hexagon with three dents

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Advances in Applied Mathematics, ISSN: 0196-8858, Vol: 82, Page: 23-57

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Tri Lai
Elsevier BV
article description
MacMahon's classical theorem on boxed plane partitions states that the generating function of the plane partitions fitting in an a×b×c box is equal to Hq(a)Hq(b)Hq(c)Hq(a+b+c)Hq(a+b)Hq(b+c)Hq(c+a), where Hq(n):=[0]q!⋅[1]q!…[n−1]q! and [n]q!:=∏i=1n(1+q+q2+…+qi−1). By viewing a boxed plane partition as a lozenge tiling of a semi-regular hexagon, MacMahon's theorem yields a natural q -enumeration of lozenge tilings of the hexagon. However, such q -enumerations do not appear often in the domain of enumeration of lozenge tilings. In this paper, we consider a new q -enumeration of lozenge tilings of a hexagon with three bowtie-shaped regions removed from three non-consecutive sides. The unweighted version of the result generalizes a problem posed by James Propp on enumeration of lozenge tilings of a hexagon of side-lengths 2 n, 2n+3, 2 n, 2n+3, 2 n, 2n+3 (in cyclic order) with the central unit triangles on the (2n+3) -sides removed. Moreover, our result also implies a q -enumeration of boxed plane partitions with certain constraints.