对称群共轭类中的排列统计
组合数学
2023-05-18 v2 概率论
摘要
我们引入了对称群上加权逆序统计的概念，并考察其在每个共轭类上的分布。我们的工作推广了若干常见排列统计的研究，包括逆序数、下降数、主指标和超位数。作为结果，我们获得了按共轭类的若干统计量的首矩的显式公式。我们还表明，当循环长度足够大时，任意排列统计量的高阶矩与共轭类无关。Fulman（J. Comb. Theory Ser. A., 1998）先前对主指标和下降数建立了这一结果。我们部分通过推广 Fulman（同上）的技术并引入排列约束的概念得到了这些结果。对于可通过对称约束实现的排列统计量，我们表明每个矩都是对称群次数的多项式。
引用
@article{arxiv.2301.00898,
title = {Permutation Statistics in Conjugacy Classes of the Symmetric Group},
author = {Jesse Campion Loth and Michael Levet and Kevin Liu and Eric Nathan Stucky and Sheila Sundaram and Mei Yin},
journal= {arXiv preprint arXiv:2301.00898},
year = {2023}
}
备注
We would also like to express our gratitude to Yan Zhuang for kindly alerting us to the arXiv paper of Hamaker and Rhoades (arXiv:2206.06567), after seeing the first version of the present paper. We also thank Zach Hamaker for taking the time to explain the results of the Hamaker--Rhoades paper and its overlap with the present work