  The partial transpose of a block matrix M is the matrix obtained by
transposing the blocks of M independently. We approach the notion of partial
transpose from a combinatorial point of view. In this perspective, we solve
some basic enumeration problems concerning the partial transpose of permutation
matrices. More specifically, we count the number of permutations matrices which
are equal to their partial transpose and the number of permutation matrices
whose partial transpose is still a permutation. We solve these problems also
when restricted to symmetric permutation matrices only.
