111 research outputs found
New proofs of determinant evaluations related to plane partitions
We give a new proof of a determinant evaluation due to Andrews, which has
been used to enumerate cyclically symmetric and descending plane partitions. We
also prove some related results, including a q-analogue of Andrews's
determinant.Comment: 25 page
Selberg integrals, Askey-Wilson polynomials and lozenge tilings of a hexagon with a triangular hole
We obtain an explicit formula for a certain weighted enumeration of lozenge
tilings of a hexagon with an arbitrary triangular hole. The complexity of our
expression depends on the distance from the hole to the center of the hexagon.
This proves and generalizes conjectures of Ciucu et al., who considered the
case of plain enumeration when the triangle is located at or very near the
center. Our proof uses Askey-Wilson polynomials as a tool to relate discrete
and continuous Selberg-type integrals.Comment: 29 pages; minor changes from v
Elliptic pfaffians and solvable lattice models
We introduce and study twelve multivariable theta functions defined by
pfaffians with elliptic function entries. We show that, when the crossing
parameter is a cubic root of unity, the domain wall partition function for the
eight-vertex-solid-on-solid model can be written as a sum of two of these
pfaffians. As a limit case, we express the domain wall partition function for
the three-colour model as a sum of two Hankel determinants. We also show that
certain solutions of the TQ-equation for the supersymmetric eight-vertex model
can be expressed in terms of elliptic pfaffians.Comment: 34 page
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