Descent polynomials
Descent polynomials
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下降多项式
DOI：
10.1016/j.disc.2019.01.034
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
B. Sagan
中科院分区：
文献类型：
--
作者：
Alexander Diaz;P. Harris;Erik Insko;Mohamed Omar;B. Sagan
设n是一个非负整数而I是一个有限正整数的集合。1915年，MacMahon证明了具有下降集I的对称群sn中的排列数是n中的一个多项式，我们称之为下降多项式。然而，这些多项式的基本性质，如它们的系数和根的描述，似乎没有在文献中研究过。最近，在2013年，Billey， burzy和Sagan证明了sn中具有峰集I的元素的个数是n乘以2的某个幂的多项式。从那时起，有大量的论文研究了这个峰值多项式的性质。本文的目的是研究下降多项式。我们会看到它和它的峰值相对物有一些有趣的相似之处。猜想和未来研究的问题散落在各处。
Let n be a nonnegative integer and I be a finite set of positive integers. In 1915, MacMahon proved that the number of permutations in the symmetric group S n with descent set I is a polynomial in n. We call this the descent polynomial. However, basic properties of these polynomials such as a description of their coefficients and roots do not seem to have been studied in the literature. Much more recently, in 2013, Billey, Burdzy, and Sagan showed that the number of elements of S n with peak set I is a polynomial in n times a certain power of two. Since then, there have been a flurry of papers investigating properties of this peak polynomial. The purpose of the present paper is to study the descent polynomial. We will see that it displays some interesting parallels with its peak relative. Conjectures and questions for future research are scattered throughout.
峰值多项式正性猜想的证明
DOI：
10.1016/j.jcta.2017.01.004
发表时间：
2017
期刊：
Series A
影响因子：
--
作者：
Diaz-Lopez, Alexander;Harris, Pamela E.;Insko, Erik;Omar, Mohamed
通讯作者：
Omar, Mohamed