Patterns in multi-dimensional permutations

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.