  Motivated by the conjecture posed by Fulmek and Krattenthaler, we provide
product formulas for the number of lozenge tilings of a hexagon with a vertical
intrusion. As a direct corollary, we obtain the product formula for the number
of plane partitions fitting inside a box with a certain restriction. We also
investigate the asymptotic behavior of the ratio between the number of lozenge
tilings of a hexagon with a vertical intrusion and that of one without it.
