We show that if a permutation statistic can be written as a linear combination of bivincular patterns, then its moments can be expressed as a linear combination of factorials with constant coefficients. This generalizes a result of Zeilberger. We use an approach of Chern, Diaconis, Kane and Rhoades, previously applied on set partitions and matchings. In addition, we give a new proof of the central limit theorem (CLT) for the number of occurrences of classical patterns, which uses a lemma of Burstein and Hasto. We give a simple interpretation of this lemma and an analogous lemma that would imply the CLT for the number of occurrences of any vincular pattern. Furthermore, we obtain explicit formulas for the moments of the descents and the minimal descents statistics. The latter is used to give a new direct proof of the fact that we do not necessarily have asymptotic normality of the number of pattern occurrences in the case of bivincular patterns. Closed forms for some of the higher moments of several popular statistics on permutations are also obtained.
翻译:我们显示,如果变异统计可以作为双视模式的线性组合写成,那么其时刻可以作为元素与恒定系数的线性组合来表达。 这概括了Zeilberger的结果。 我们使用切恩、迪亚科尼斯、凯恩和罗德斯的方法, 先前在设定分区和匹配上应用过。 此外, 我们给出新的证据, 证明古典模式的发生次数的中央限值理论( CLT), 古典模式使用布尔斯坦和哈斯托的利玛。 我们简单解释这一元素和类似的利玛, 表示任何隐性模式发生次数的CLT。 此外, 我们获得关于世系和最小血统统计的明确公式。 后者用来提供新的直接证据, 证明我们不一定在双视模式中出现模式的规律性常态。 某些较高级的流行统计也以封闭的形式获得。