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

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• 作者：Tri Lai ^{tlai@umn.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05A15 ; 05C30 ; 05C70
• 刊名：Advances in Applied Mathematics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：82
• 期：Complete
• 页码：23-57
• 全文大小：952 K

文摘

MacMahon's classical theorem on boxed plane partitions states that the generating function of the plane partitions fitting in an 885816300586&_mathId=si1.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=e8e1679310ca297e0101c78315b4a909" title="Click to view the MathML source">a×b×c$a\times b\times c$ box is equal to where 885816300586&_mathId=si3.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=186a8673c0da87638b7952cec4b9a6b7" title="Click to view the MathML source">H_q(n):=[0]_q!⋅[1]_q!…[n−1]_q!${\mathrm{H}}_q(n):={[0]}_q!\cdot {[1]}_q!\dots {[n-1]}_q!$ and 885816300586&_mathId=si4.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=a611a5671c59987b3577d0717b491546">885816300586-si4.gif">${[n]}_q!:={\prod }_{i=1}^n(1+q+q^2+\dots +q^{i-1})$. 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  , 885816300586&_mathId=si5.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=e24e41ee03bc80f7bf53ba74222971c7" title="Click to view the MathML source">2n+3$2n+3$, 2n  , 885816300586&_mathId=si5.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=e24e41ee03bc80f7bf53ba74222971c7" title="Click to view the MathML source">2n+3$2n+3$, 2n  , 885816300586&_mathId=si5.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=e24e41ee03bc80f7bf53ba74222971c7" title="Click to view the MathML source">2n+3$2n+3$ (in cyclic order) with the central unit triangles on the 885816300586&_mathId=si7.gif&_user=111111111&_pii=S0196885816300586&_rdoc=1&_issn=01968858&md5=9abe68a4265a2c4e2e84fb40505a96e6" title="Click to view the MathML source">(2n+3)$(2n+3)$-sides removed. Moreover, our result also implies a q-enumeration of boxed plane partitions with certain constraints.