Plane permutations and applications to a result of Zagier-Stanley and distances of permutations
Ricky X. F. Chen, Christian M. Reidys
备注：
To appear in SIAM J. Discrete Math. Considering the scope of the journal, the content in arXiv:1502.07971 "A simple framework on sorting permutations" was included
In this paper, we introduce plane permutations, i.e. pairs $\mathfrak{p}=(s,π)$ where $s$ is an $n$-cycle and $π$ is an arbitrary permutation, represented as a two-row array. Accordingly a plane permutation gives rise to three distinct permutations: the permutation induced by the upper horizontal ($s$), the vertical $π$) and the diagonal ($D_{\mathfrak{p}}$) of the array. The latter can also be viewed as the three permutations of a hypermap. In particular, a map corresponds to a plane permutation, in which the diagonal is a fixed point-free involution. We study the transposition action on plane permutations obtained by permuting their diagonal-blocks. We establish basic properties of plane permutations and study transpositions and exceedances and derive various enumerative results. In particular, we prove a recurrence for the number of plane permutations having a fixed diagonal and $k$ cycles in the vertical, generalizing Chapuy's recursion for maps filtered by the genus. As applications of this framework, we present a combinatorial proof of a result of Zagier and Stanley, on the number of $n$-cycles $ω$, for which the product $ω(1~2~\cdots ~n)$ has exactly $k$ cycles. Furthermore, we integrate studies on the transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. Plane permutations allow us to generalize and recover various lower bounds for transposition and block-interchange distances and to connect reversals with block-interchanges.
Asymptotics of principal evaluations of Schubert polynomials for layered permutations
Department of Mathematics and Statistics, UMass, Amherst, MA（马萨诸塞大学阿默斯特分校数学与统计系）；Department of Mathematics, UCLA, Los Angeles, CA（加州大学洛杉矶分校数学系）；Institute of Advanced Studies, Princeton, NJ（普林斯顿高等研究院；宾夕法尼亚大学数学系）；Department of Mathematics, UPenn, Philadelphia, PA
Generalized Stirling permutations, families of increasing trees and urn models
Bona [2007+]研究了Gessel和Stanley [1978]引入的Stirling排列类中上升、平稳和下降的分布。最近，Janson [2008+]证明了Stirling置换和平面递归树之间的联系，并证明了Bona考虑的参数的联合正规律。这里我们将考虑广义Stirling置换，推广了Bona和Janson的早期结果，并将它们与某些广义平面递归树族以及$(k+1)$ -元增树联系起来。我们还给出了两个不同的双射之间的某些家庭的增长树，这两个作为一个特殊情况的双射之间的三元增长树和平面递归树。为了描述感兴趣的参数的（渐近）行为，我们研究了三个（广义）Polya瓮模型使用各种方法。
