TY - JOUR

T1 - A q-enumeration of lozenge tilings of a hexagon with three dents

AU - Lai, Tri

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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 toi=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 2n, 2n+3, 2n, 2n+3, 2n, 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.

AB - 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 toi=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 2n, 2n+3, 2n, 2n+3, 2n, 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.

KW - Graphical condensation

KW - Lozenge tilings

KW - Perfect matchings

KW - Plane partitions

UR - http://www.scopus.com/inward/record.url?scp=84979581515&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84979581515&partnerID=8YFLogxK

U2 - 10.1016/j.aam.2016.07.002

DO - 10.1016/j.aam.2016.07.002

M3 - Article

AN - SCOPUS:84979581515

VL - 82

SP - 23

EP - 57

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

ER -