Max-tree for d-permutations and pattern avoidance

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 $2<sup>{d-1}-$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.

• The paper generalizes the max-tree construction to d-permutations, mapping admissible classes defined by compatible orders bijectively to 2^(d−1)-ary trees.
• The paper proves that d-permutations avoiding (12) and (231) correspond bijectively to d-ary trees, yielding the enumeration formula 1/(dn+1) binom(dn+1,n).
• The result resolves a conjecture by Bonichon and Morel while extending the classical Catalan correspondence for 231-avoiding permutations to higher dimensions and identifying open questions about other axes and twin trees.

Background and motivation

A $d$-permutation of size $n$ is a tuple of $d-1$ permutations of $[n]$, equivalently a set of $n$ points in a $d$-dimensional grid with exactly one point per hyperplane perpendicular to each axis. Pattern avoidance for such objects admits several inequivalent definitions; this paper adopts the projection-based notion introduced by Bonichon and Morel, in which a $d$-permutation contains a $d'$-dimensional pattern if some direct projection onto $d'$ coordinates, restricted to an increasing set of indices, is order-isomorphic to the pattern.

The point of departure is the classical max-tree construction. For an ordinary permutation viewed as a planar point set, the max-tree has as root the point with maximal value; deleting it splits the remaining points into left and right subpermutations, which recursively form the two subtrees. This map is surjective onto binary trees but not injective — for instance, 41523 and 43512 share the same max-tree — yet it becomes a bijection when restricted to 231-avoiding permutations, recovering the Catalan enumeration familiar from Knuth's analysis of stack-sortable permutations. Dually, min-tree/max-tree pairs (twin binary trees) are in bijection with Baxter permutations via Dulucq and Guibert.

The paper generalizes this construction to arbitrary dimension and uses it to settle a conjecture of Bonichon and Morel on the enumeration of a specific pattern-avoiding class of $d$-permutations.

The max-tree of a $n$0-permutation

Fix the last coordinate axis $n$1. The maximal point $n$2 of a $n$3-permutation $n$4 becomes the root, and the remaining points are partitioned according to their direction relative to the root among the $n$5 directions negative with respect to $n$6 (the set $n$7). Recursing yields a $n$8-ary tree whose children at each node are labeled by these directions. In concrete terms, the max-tree of a 3-permutation is the quadtree of its underlying 2D point set, and that of a 4-permutation is the octree of a 3D point set, connecting the construction to the classical hyperoctree data structures of Finkel–Bentley and Yau–Srihari.

As in dimension two, the induced map $n$9 is surjective but not injective. The paper's first structural contribution identifies exactly which fibers become singletons under suitable restrictions.

Admissible classes and the general bijection

The key notion is a compatible set of total orders $d-1$0 on $d-1$1: each order $d-1$2 must place every direction with a $d-1$3 in coordinate $d-1$4 after every direction with a $d-1$5 there. A $d-1$6-permutation is admissible with respect to $d-1$7 if, at every internal node of its max-tree, the subpermutations hanging from the node appear as ordered blocks along each coordinate axis, ordered as prescribed by the corresponding $d-1$8.

Main theorem (general form). For any compatible set of total orders $d-1$9, the admissible class $[n]$0 is in bijection with the set of $[n]$1-ary trees with $[n]$2 internal nodes.

The proof proceeds by showing $[n]$3 is surjective (an induction assembling subtrees into block-ordered permutations, using compatibility of $[n]$4 to guarantee the block arrangement exists) and injective (if two admissible permutations map to the same tree but differ on some pair of nodes, the ancestor/sibling case analysis forces one of them to violate the block ordering). A corollary extends the result to non-power-of-two arities: restricting to a subset $[n]$5 of $[n]$6 directions with compatible orders on $[n]$7 gives a bijection between the corresponding admissible class and $[n]$8-ary trees, obtained by pruning leaves of directions outside $[n]$9.

This theorem already implies that every admissible class is counted by Fuss–Catalan numbers of the appropriate arity, and it frames the pattern-avoidance result below as a special case rather than an isolated fact.

Pattern avoidance and $n$0-ary trees

Let $n$1 denote the class of $n$2-permutations avoiding the pattern $n$3 of size two and dimension three together with the pattern $n$4 of size three and dimension two, under the Bonichon–Morel avoidance definition. Define $n$5, where $n$6 has $n$7 leading $n$8 signs followed by $n$9 signs, equipped with the single compatible order placing $d$0 iff $d$1.

Main theorem. $d$2 is in bijection with the set of $d$3-ary trees with $d$4 internal nodes, and consequently

$d$5

The proof establishes three implications. First, avoiding $d$6 forces every direction between two points to lie in $d$7 (any direction with interleaved signs produces a $d$8 occurrence), so all internal nodes of the max-tree carry directions in $d$9. Second, avoiding both patterns forces admissibility with respect to $d$0: the argument shows that any violation of the block ordering, combined with the direction constraints, generates either a $d$1 or a $d$2 occurrence. Third, conversely, any $d$3 or $d$4 occurrence in a $d$5-avoiding permutation violates admissibility; this last step requires an exhaustive case analysis over the possible configurations of two or three internal nodes in the max-tree (ancestor–descendant relations versus shared parents), carried out in the appendix with labeled configuration diagrams.

Two consequences deserve emphasis. The enumeration formula resolves the conjecture posed by Bonichon and Morel and extended in Muller's thesis, in the affirmative. Moreover, the bijection extends the classical Knuth correspondence between 231-avoiding permutations and binary trees to all dimensions, while retaining a remarkably simple forbidden-pattern characterization: only two patterns, one of size two/dimension three and one of size three/dimension two.

The paper also notes a contrast with dimension two that functions as a structural observation: in the plane, all permutation classes avoiding a single pattern of length three are equinumerous (all Catalan-counted). In higher dimensions, among the classes examined, only $d$6 appears to share the Fuss–Catalan enumeration of $d$7; no Wilf-equivalence theory explaining this is developed here.

Limitations and open questions

Several caveats bound the scope of the results. The bijection is tied to the max-tree taken with respect to the last axis specifically; whether analogous pattern-characterized classes arise from constructions based on other axes is left open. Likewise, the restriction to the direction subset $d$8 is what reduces the arity from $d$9 to $d'$0; the existence of a more general pattern-avoiding class in bijection with full $d'$1-ary trees remains unproven. Finally, the higher-dimensional analogue of the twin-tree/Baxter-permutation correspondence is untouched: the paper defines min-trees dually but does not identify structural conditions on (min-tree, max-tree) pairs of $d'$2-permutations, nor permutation classes in bijection with such pairs. These three questions are stated explicitly by the author as follow-up problems.

Conclusion

The paper constructs a natural generalization of the max-tree to $d'$3-permutations, proves that admissible classes defined by compatible total orders on the negative directions are in bijection with $d'$4-ary trees, and specializes this framework to show that $d'$5-permutations avoiding $d'$6 and $d'$7 are enumerated by $d'$8, thereby resolving a standing conjecture. The result simultaneously generalizes the classical Catalan bijection for 231-avoiding permutations and adds to a growing body of bijective results connecting high-dimensional pattern avoidance to triangle solitaire and higher-dimensional floorplans.