Separable d-Permutations and Guillotine Partitions

Abstract

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 separable 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 several results on patterns in permutations.

Dushnik-Miller dimension of contact systems of d-dimensional boxes

“…, p d−1 ) that is contained in at least d + 1 of these (d − 1)-boxes and also such that p ∈ A ∩ B. Let these (d − 1)-boxes be A, B, H (1) , . .…”

“…. , 1) . By Lemma 1, there exists a box F that contains the point p and touches A in dimension d. This means that…”

“…Definitions are provided in the following. Note that both, the objects (graphs or simplicial complexes) with Dushnik-Miller dimension greater than three [4,7,9], and the systems of interior disjoint axis parallel boxes in R d [2,3,1], are difficult to handle but are raising interest in the community. The proof of our result generalizes H. Zhang's idea for constructing Schnyder woods in tilings of R 2 , and relies on some properties of tilings that are of independent interest.…”

A Note on Flips in Diagonal Rectangulations

“…Similarly, guillotine partitions of a rectangle into n rectangles, obtained by recursive splitting with a horizontal or vertical cut, can easily be seen to be in one-to-one correspondence with t3142, 2413u-avoiding permutations, called separable permutations [6], which are counted by the Schröder numbers (OEIS A006318). This has recently been generalized to separable d-permutations and higher-dimensional guillotine partitions [5].…”

“…Table 1 lists the known bijections between families of pattern-avoiding permutations and rectangulations. (i) Guillotine partitions of 2 d´1 -dimensional boxes [5] Tab. 1: Known bijections between families of pattern-avoiding permutations and rectangulations.…”