On d-Permutations and Pattern Avoidance Classes

IF 0.6 4区数学 Q4 MATHEMATICS, APPLIED

Annals of Combinatorics Pub Date : 2024-05-18 DOI:10.1007/s00026-024-00695-1

Nathan Sun

引用次数: 0

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annals of Combinatorics 数学-应用数学

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board. The scope of Annals of Combinatorics is covered by the following three tracks: Algebraic Combinatorics: Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices Analytic and Algorithmic Combinatorics: Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms Graphs and Matroids: Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches