Separable d-permutations and guillotine partitions

TL;DR

This paper characterizes and enumerates separable multidimensional permutations using forbidden patterns and explores their connection to guillotine partitions, providing new insights and bijections in combinatorial structures.

Contribution

It introduces a characterization of separable multidimensional permutations and establishes a bijection with guillotine partitions, advancing understanding of their combinatorial properties.

Findings

Enumeration formulas for separable multidimensional permutations

Bijection between permutations and guillotine partitions

Results on pattern avoidance in permutations

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 $d$-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.