Shuffle-compatible permutation statistics
Shuffle-compatible permutation statistics
复制标题
兼容随机播放的排列统计
DOI：
10.1016/j.aim.2018.05.003
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发表时间：
2017
影响因子：
1.7
通讯作者：
Zhuang Yan
中科院分区：
文献类型：
--
作者：
I. Gessel;Zhuang Yan
自从理查德·斯坦利的早期工作以来，人们已经观察到一些排列统计量对于排列的洗牌具有显着的性质。我们正式这个概念的洗牌兼容的置换统计，并引入洗牌代数的洗牌兼容的置换统计，编码的统计分布在洗牌的置换。本文发展了一个理论的洗牌兼容性下降的统计数据，只依赖于下降集和长度，这有密切的联系，理论ofP-分区，拟对称函数，和非交换对称函数。我们使用我们的框架来证明，许多下降统计是洗牌兼容的，并给出明确的描述，他们的洗牌代数，从而统一过去的结果斯坦利，Gessel，Stembridge，阿吉亚尔-Bergeron-Nyman和彼得森。
Since the early work of Richard Stanley, it has been observed that several permutation statistics have a remarkable property with respect to shuffles of permutations. We formalize this notion of a shuffle-compatible permutation statistic and introduce the shuffle algebra of a shuffle-compatible permutation statistic, which encodes the distribution of the statistic over shuffles of permutations. This paper develops a theory of shuffle-compatibility for descent statistics—statistics that depend only on the descent set and length—which has close connections to the theory ofP-partitions, quasisymmetric functions, and noncommutative symmetric functions. We use our framework to prove that many descent statistics are shuffle-compatible and to give explicit descriptions of their shuffle algebras, thus unifying past results of Stanley, Gessel, Stembridge, Aguiar–Bergeron–Nyman, and Petersen.