  We have derived the perimeter generating function of a model of punctured
staircase polygons in which the internal staircase polygon is rotated by a
90degree angle with respect to the outer staircase polygon. In one approach we
calculated a long series expansion for the problem and found that all the terms
in the generating function can be reproduced from a linear Fuchsian
differential equation of order 4. We then solved this ODE and found a closed
form expression for the generating function. This is a highly unusual and most
fortuitous result since ODEs of such high order very rarely permit a closed
form solution. In a second approach we proved the result for the generating
function exactly using combinatorial arguments. This latter solution allows
many generalisations including to models with other types of punctures and to a
model with any fixed number of nested rotated staircase punctures.
