Max-tree for d-permutations and pattern avoidance

TL;DR

This paper introduces a generalized max-tree mapping for higher-dimensional permutations, establishing a bijection with d-ary trees for certain pattern-avoiding classes, thus resolving a conjecture in the field.

Contribution

It generalizes the max-tree construction to d-permutations and proves a bijection with d-ary trees for specific pattern-avoiding classes, confirming a prior conjecture.

Findings

Mapping from d-permutations to d-ary trees is bijective for certain pattern-avoiding classes.

Resolves a conjecture about enumeration of pattern avoiding d-permutations.

Generalizes classical max-tree construction to higher dimensions.

Abstract

Higher dimensional permutations are tuples of d-1 permutations that can be identified with a point set in a d-dimensional grid. In N. Bonichon and P.-J. Morel, {\it J. Integer Sequences} 25 (2022), several conjectures regarding the enumeration of pattern avoiding d-permutations were stated. In this paper, we consider a mapping from d-permutations to $2^{d-1}-$ary trees that naturally generalizes the classical max-tree construction for permutations. We then show that, when restricted to d-permutations avoiding (21,12) and 231, this mapping yields a bijection with d-ary trees. This result resolves one of the conjectures of Bonichon and Morel.