  In their paper about a dual of MacMahon's classical theorem on plane
partitions, Ciucu and Krattenthaler proved a closed form product formula for
the tiling number of a hexagon with a "shamrock", a union of four adjacent
triangles, removed in the center (Proc. Natl. Acad. Sci. USA 2013). Lai later
presented a $q$-enumeration for lozenge tilings of a hexagon with a shamrock
removed from the boundary (European J. Combin. 2017). It appears that the above
are the only two positions of the shamrock hole that yield nice tiling
enumerations. In this paper, we show that in the case of symmetric hexagons, we
always have a simple product formula for the number of tilings when removing a
shamrock at any position along the symmetry axis. Our result also generalizes
Eisenk\"olbl's related work about lozenge tilings of a hexagon with two unit
triangles missing on the symmetry axis (Electron. J. Combin. 1999).
