# Enumeration of Lozenge Tilings of a Hexagon with Holes on Boundary

Tuesday, February 3, 2015 - 2:00pm - 3:00pm

Lind 305

Tri Lai (University of Minnesota, Twin Cities)

MacMahon's classical theorem on the number of plane partitions that fit in a given box is equivalent to fact that the number of lozenge tilings of a semi-regular hexagon of side-lengths $a,b,c,a,b,c$ (in cyclic order) on the triangular lattice is equal to

\[\frac{H(a)H(b)H(c)H(a+b+c)}{H(a+b)H(b+c)H(c+a)},\]

where the hyperfactorial function $H(n)$ is defined by

\[H(n):=0!1!\dots(n-1)!.\]

We generalize MacMahon's theorem by enumerating the lozenge tilings of a hexagon with holes on its boundary. In addition, we investigate a $q$-enumeration of plane partitions that fit in a special box consisting of several connected rooms.

\[\frac{H(a)H(b)H(c)H(a+b+c)}{H(a+b)H(b+c)H(c+a)},\]

where the hyperfactorial function $H(n)$ is defined by

\[H(n):=0!1!\dots(n-1)!.\]

We generalize MacMahon's theorem by enumerating the lozenge tilings of a hexagon with holes on its boundary. In addition, we investigate a $q$-enumeration of plane partitions that fit in a special box consisting of several connected rooms.