Baxter $d$-permutations and other pattern avoiding classes

TL;DR

This paper explores multidimensional permutations, classifies small pattern avoiding classes, establishes bijections, and generalizes Baxter permutations to higher dimensions with vincular pattern characterizations.

Contribution

It introduces the concept of Baxter $d$-permutations and extends pattern avoidance theory into multidimensional settings.

Findings

Complete enumeration of small pattern avoiding classes

Bijections for certain classes

Characterization of Baxter $d$-permutations via vincular patterns

Abstract

A permutation of size $n$ can be identified to its diagram in which there is exactly one point per row and column in the grid $[n]^2$. In this paper we consider multidimensional permutations (or $d$-permutations), which are identified to their diagrams on the grid $[n]^d$ in which there is exactly one point per hyperplane $x_i=j$ for $i\in[d]$ and $j\in[n]$. We first investigate exhaustively all small pattern avoiding classes. We provide some bijection to enumerate some of these classes and we propose some conjectures for others. We then give a generalization of well-studied Baxter permutations into this multidimensional setting. In addition, we provide a vincular pattern avoidance characterization of Baxter $d$-permutations.