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关于避免集合划分的多模式

On Multiple Pattern Avoiding Set Partitions

Vít Jelínek, Toufik Mansour, Mark Shattuck

备注：
37 pages. Corrected a typo
发表信息：
Advances in Applied Mathematics, 50(2) (2013), 292-326
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摘要：

我们研究类集分区确定的避免多个模式，应用自然的概念分区遏制已被引入萨根。我们说两个模式集S和T是等价的，如果对于每个n，避免S的所有成员的大小为n的分区的数量与避免T的所有成员的分区的数量相同。我们的目标是分类的等价类之间的两个元素的模式集的几个一般类型。首先，我们关注模式对{σ，τ}，其中σ是具有至少两个不同符号的大小为3的模式，τ是避免σ的大小为k的任意模式。我们表明，这种类型的模式对确定少量的等价类，特别是，类的平均指数大小在k。我们提供了一个（次指数）的上界的等价类的数量，并提供了一个明确的公式，所有这些避免类的生成函数，表明在所有情况下，这个生成函数是合理的。接下来，我们研究避免一对{1212，τ}形式的模式的划分，其中τ是任意模式。注意，避免分区1212恰好是非交叉分区。我们为这种模式对提供了几个通用的等价准则，并证明了当τ的大小不超过6时，这些准则可以解释所观察到的所有等价性。在本文的最后一部分，我们对所有对{σ，τ}的等价类进行了完全分类，其中σ和τ的大小为4。

英文摘要：

We study classes of set partitions determined by the avoidance of multiple patterns, applying a natural notion of partition containment that has been introduced by Sagan. We say that two sets S and T of patterns are equivalent if for each n, the number of partitions of size n avoiding all the members of S is the same as the number of those that avoid all the members of T. Our goal is to classify the equivalence classes among two-element pattern sets of several general types. First, we focus on pairs of patterns {σ,τ}, where σ is a pattern of size three with at least two distinct symbols and τ is an arbitrary pattern of size k that avoids σ. We show that pattern-pairs of this type determine a small number of equivalence classes; in particular, the classes have on average exponential size in k. We provide a (sub-exponential) upper bound for the number of equivalence classes, and provide an explicit formula for the generating function of all such avoidance classes, showing that in all cases this generating function is rational. Next, we study partitions avoiding a pair of patterns of the form {1212,τ}, where τ is an arbitrary pattern. Note that partitions avoiding 1212 are exactly the non-crossing partitions. We provide several general equivalence criteria for pattern pairs of this type, and show that these criteria account for all the equivalences observed when τ has size at most six. In the last part of the paper, we perform a full classification of the equivalence classes of all the pairs {σ,τ}, where σ and τ have size four.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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