Further determinants with the averaging property of Andrews-Burge

The determinants which arise in the enumeration of symmetry classes of plane partitions are difficult to evaluate. Recently, Andrews and Burge have shown that the determinant of one of these classes has a surprising property, which we call the averaging property. We obtain a class of determinants under reasonably general conditions which also possess this property. This class includes the Andrews and Burge example. We also consider conditions under which determinants in this class can be evaluated.