模式回避单射词复形

中文摘要 AI 辅助

单射词复形是一种出现在多个不同领域中的胞腔复形。它应用于证明同调稳定性以及群上同调的研究，并且与随机到随机马尔可夫链密切相关。该复形最早由Farmer研究，他证明了它具有一个由最高维球面楔和构成的同调。后来，Björner和Wachs建立了它的可壳性，Reiner和Webb揭示了其$S_n$-模结构，并在此过程中观察到其最高同调群的秩是第$n$个错排数。我们通过固定一个置换模式$\sigma$，并仅考虑字母表$\{1,2,\dots,n\}$中回避$\sigma$的单射词，引入了单射词复形的自然子复形。我们证明了如果$\sigma$以其最大或最小字母开头或结尾，则此类模式回避复形是可壳的，并为回避此类模式的复形构造了同调基。对于长度为3的模式（所有这些模式都具有此性质），我们证明了所得复形的最高同调的秩是一个Riordan数。长度为4的模式中除四个外也都具有此性质，对于其余四个模式中的两个，我们使用另一种方法建立了可壳性。我们还引入了一种利用枚举组合学来证明可壳性的技术，并将其应用于可分单射词复形，从而在此情形下推导出可壳性。在此过程中，我们给出了完整单射词复形以及我们证明可壳的每个子复形的所有$h$-数的组合公式。反过来，我们利用模式回避单射词复形的可壳性推导出模式回避置换的新的精细计数公式。

英文摘要

The complex of injective words is a cell complex that arises in a number of different areas. It has applications to proving homological stability and to the study of group cohomology, and it is closely related to the random-to-random Markov chain. This complex was first studied by Farmer, who proved it has the homology of a wedge of top-dimensional spheres. Later, Björner and Wachs established its shellability, and Reiner and Webb uncovered its $S_n$-module structure, observing in the process that the rank of its top homology group is the $n$th derangement number. We introduce natural subcomplexes of the complex of injective words by fixing a permutation pattern $σ$ and considering only those injective words in the alphabet $\{1,2,\dots,n\} $ that avoid $σ$. We prove that such pattern-avoiding complexes are shellable if $σ$ begins or ends with its largest or smallest letter, and we construct homology bases for the complexes avoiding such patterns. For patterns of length 3, all of which have this property, we show that the rank of the top homology of the resulting complex is a Riordan number. All but four patterns of length 4 also have this property, and for two of the remaining four patterns, we establish shellability using a different method. We also introduce a technique to use enumerative combinatorics to prove shellability, and we apply it to the complex of separable injective words, thereby deducing shellability in this case. Along the way, we give a combinatorial formula for all of the $h$-numbers in the full complex of injective words as well as for each of the subcomplexes which we prove are shellable. Going in the other direction, we use shellability of complexes of pattern-avoiding injective words to deduce new refined counting formulas for pattern-avoiding permutations.

发表机构

• Dartmouth College（达特茅斯学院）
• University of Oregon（俄勒冈大学）

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