Flagged (P,ρ) -partitions
Flagged (P,ρ) -partitions
复制标题
标记 (P,Ï) - 分区
DOI：
10.1016/j.ejc.2020.103085
复制
发表时间：
2020
影响因子：
1
通讯作者：
Bergeron, Nantel
中科院分区：
文献类型：
--
作者：
Assaf, Sami;Bergeron, Nantel
引入了（P，ρ）-划分理论，它依赖于偏序集P和从P到正整数的映射ρ.（P，ρ）-划分的生成函数FP，ρ是一个多项式，当ρ的象趋于无穷大时，它趋于P-划分的Stanley生成函数。类似于Stanley关于P-划分的基本定理，我们证明了（P，ρ）-划分的集合分解为（L，ρ）-划分的不交并，其中L在P的线性扩张的集合上运行。因此，我们引入了标记（P，ρ）-划分的概念，并证明了线性阶L的标记（L，ρ）-划分的所有FL，ρ的集合正是第一作者和Searles引入的多项式环的基本滑动基。我们的主要定理表明，任何标记（P，ρ）-分拆的生成函数FP，ρ都是g滑动多项式的正整数线性组合.作为应用，我们给出了一个新的证明的积极性的幻灯片产品，激励我们的命名，我们也证明了标记Schur函数是幻灯片积极的。
We introduce the theory of (P, ρ)-partitions, depending on a poset P and a map ρ from P to positive integers. The generating function F P, ρ of (P, ρ)-partitions is a polynomial that, when the images of ρ tend to infinity, tends to Stanley’s generating function of P-partitions. Analogous to Stanley’s fundamental theorem for P-partitions, we show the set of (P, ρ)-partitions decomposes as a disjoint union of (L, ρ)-partitions where L runs over the set of linear extensions of P. In this more general context, the set of all F L, ρ for linear orders L over determines a basis of polynomials. We thus introduce the notion of flagged (P, ρ)-partitions, and we prove the set of all F L, ρ for flagged (L, ρ)-partitions for linear orders L is precisely the fundamental slide basis of the polynomial ring, introduced by the first author and Searles. Our main theorem shows that any generating function F P, ρ of flagged (P, ρ)-partitions is a positive integer linear combination g slide polynomials. As applications, we give a new proof of positivity of the slide product and, motivating our nomenclature, we also prove flagged Schur functions are slide positive.