论文标题
列出置换的多个公式
Plethystic formulas for permutation enumeration
论文作者
论文摘要
我们证明了各种置换统计量的分布的几个通用公式。我们的公式涉及准对称生成函数上的某些多种替代，我们考虑的排列统计数据包括下降数，峰值数，左峰号和上下运行的数量。我们将这些结果应用于循环排列，相关性和毁灭性，更普遍地将这些结果用于得出公式，以通过上述统计数据与固定点的数量以及循环类型共同计数上述统计数据进行计数。许多已知的公式被恢复为我们结果的特殊情况，包括Désarménien-Foata，Gessel-Reutenauer，Stembridge，Fulman，Fulman，Petersen，Diaconis-Fulman-Holmes，Zhuang和Zhuang和Athanasiadis的公式。
We prove several general formulas for the distributions of various permutation statistics over any set of permutations whose quasisymmetric generating function is a symmetric function. Our formulas involve certain kinds of plethystic substitutions on quasisymmetric generating functions, and the permutation statistics we consider include the descent number, peak number, left peak number, and the number of up-down runs. We apply these results to cyclic permutations, involutions, and derangements, and more generally, to derive formulas for counting all permutations by the above statistics jointly with the number of fixed points and jointly with cycle type. A number of known formulas are recovered as special cases of our results, including formulas of Désarménien-Foata, Gessel-Reutenauer, Stembridge, Fulman, Petersen, Diaconis-Fulman-Holmes, Zhuang, and Athanasiadis.
