Cut equivalence of dd-dimensional guillotine partitions

A guillotine partition   of a dd-dimensional axis-aligned box BB is a recursive partition of BB by axis-aligned hyperplane cuts. The size of a guillotine partition is the number of boxes it contains. Two guillotine partitions are box-equivalent if their boxes satisfy compatible order relations with respect to the axes. (In many works, box-equivalent guillotine partitions are considered identical.) In the present work we define cut-equivalence   of guillotine partitions, derived in a similar way from order relations of cuts. We prove structural properties related to these kinds of equivalence, and enumerate cut-equivalence classes of dd-dimensional guillotine partitions of size  nn.