MacMahon's classical theorem on boxed plane partitions states that the generating function of the plane partitions fitting in an 0586&_mathId=si1.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=e8e1679310ca297e0101c78315b4a909" title="Click to view the MathML source">a×b×c box is equal to
where 0586&_mathId=si3.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=186a8673c0da87638b7952cec4b9a6b7" title="Click to view the MathML source">Hq(n):=[0]q!⋅[1]q!…[n−1]q! and 0586&_mathId=si4.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=a611a5671c59987b3577d0717b491546">0586-si4.gif">. 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 , 0586&_mathId=si5.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=e24e41ee03bc80f7bf53ba74222971c7" title="Click to view the MathML source">2n+3, 2n , 0586&_mathId=si5.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=e24e41ee03bc80f7bf53ba74222971c7" title="Click to view the MathML source">2n+3, 2n , 0586&_mathId=si5.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=e24e41ee03bc80f7bf53ba74222971c7" title="Click to view the MathML source">2n+3 (in cyclic order) with the central unit triangles on the 0586&_mathId=si7.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=9abe68a4265a2c4e2e84fb40505a96e6" title="Click to view the MathML source">(2n+3)-sides removed. Moreover, our result also implies a q-enumeration of boxed plane partitions with certain constraints.