  We characterize separable multidimensional permutations in terms of forbidden
patterns and enumerate them by means of generating function, recursive formula
and explicit formula. We find a connection between multidimensional
permutations and guillotine partitions of a box. In particular, a bijection
between $d$-dimensional permutations and guillotine partitions of a
$2^{d-1}$-dimensional box is constructed. We also study enumerating problems
related to guillotine partitions under certain restrictions revealing
connections to other combinatorial structures. This allows us to obtain results
on avoided patterns in permutations.

Combinatorics