On $d$-permutations and Pattern Avoidance Classes

Abstract

Bonichon and Morel first introduced d-permutations in their study of multidimensional permutations. Such permutations are represented by their diagrams on [n] d such that there exists exactly one point per hyperplane x i that satisfies x i = j for i ∈ [d] and j ∈ [n]. Bonichon and Morel previously enumerated 3-permutations avoiding small patterns, and we extend their results by first proving four conjectures, which exhaustively enumerate 3-permutations avoiding any two fixed patterns of size 3. Further, we relate 3-permutation avoidance classes with their respective recurrence relations, which lead to interesting combinatorial properties of these 3-permutation avoidance classes. In particular, we show a recurrence relation for 3-permutations avoiding the patterns 132 and 213, which contributes a new sequence to the OEIS database. We then extend our results to completely enumerate 3-permutations avoiding three patterns of size 3.