Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8904747 | Advances in Mathematics | 2018 | 57 Pages |
Abstract
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 of P-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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Ira M. Gessel, Yan Zhuang,