Patterns in Multi-dimensional Permutations

TL;DR

This paper introduces a unified framework for multi-dimensional permutation patterns, connecting them to known combinatorial sequences and providing new interpretations for sequences like Springer numbers.

Contribution

It extends permutation pattern theory to higher dimensions, defines levels in multi-dimensional permutations, and links these to OEIS sequences with novel combinatorial interpretations.

Findings

Connected multi-dimensional permutations to OEIS sequences.

Provided a combinatorial interpretation for Springer numbers.

Unified various combinatorial objects under a common framework.

Abstract

In this paper, we propose a general framework that extends the theory of permutation patterns to higher dimensions and unifies several combinatorial objects studied in the literature. Our approach involves introducing the concept of a "level" for an element in a multi-dimensional permutation, which can be defined in multiple ways. We consider two natural definitions of a level, each establishing connections to other combinatorial sequences found in the Online Encyclopedia of Integer Sequences (OEIS). Our framework allows us to offer combinatorial interpretations for various sequences found in the OEIS, many of which previously lacked such interpretations. As a notable example, we introduce an elegant combinatorial interpretation for the Springer numbers: they count weakly increasing 3-dimensional permutations under the definition of levels determined by maximal entries.