Spanning Trees of the Generalised Union Jack Lattice

The Union Jack lattice UJL(n, m) with toroidal boundary condition can be obtained from an n x m square lattice with toroidal boundary condition by inserting a new vertex v(f) to each face f and adding four edges (v(f), u(i)(f)), where u(1)(f), u(2)(f), u(3)(f), and u(4)(f) are four vertices on the boundary of f. The Union Jack lattice has been studied extensively by statistical physicists. In this article, we consider the problem of enumeration of spanning trees of the so-called generalised Union Jack lattice UDn, which is obtained from the Aztec diamond AD(n)(t) of order n with toroidal boundary condition by inserting a new vertex v(f) to each face f and adding four edges (v(f), u(i)(f)), where u(1)(f), u(2)(f), u(3)(f) and u(4)(f) are four vertices on the boundary of f.