On d-Permutations and Pattern Avoidance Classes

Pub Date : 2024-05-18 DOI:10.1007/s00026-024-00695-1
Nathan Sun
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Abstract

Multidimensional permutations, or d-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 \(i \in [d]\) and \(j \in [n]\). 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. We further provide a enumerative result relating 3-permutation avoidance classes with their respective recurrence relations. 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.

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关于 d-Permutations 和模式规避类
多维排列或 d-permutations 由它们在 \([n]^d\) 上的图来表示,对于 \(i 在 [d]\) 和 \(j 在 [n]\) 而言,每个超平面 \(x_i\) 恰好存在一个满足 \(x_i= j\) 的点。Bonichon 和 Morel 以前枚举过避免小图案的 3 次变,我们通过首先证明四个猜想来扩展他们的结果,这四个猜想详尽地枚举了避免大小为 3 的任意两个固定图案的 3 次变。我们还进一步提供了一个枚举结果,它将 3 可变避免类与它们各自的递推关系联系起来。特别是,我们展示了避开 132 和 213 图案的 3 次迭代的递推关系,这为 OEIS 数据库贡献了一个新序列。然后,我们将结果扩展到完全枚举避开三个大小为 3 的图案的 3 次变异。
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