Parabolic Catalan numbers count efficient inputs for Gessel-Viennot flagged Schur function determinant
Parabolic Catalan numbers count efficient inputs for Gessel-Viennot flagged Schur function determinant
复制标题
抛物线加泰罗尼亚数计算 Gessel-Viennot 标记的 Schur 函数行列式的有效输入
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
M. J. Willis
中科院分区：
文献类型：
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作者：
Robert A. Proctor;M. J. Willis
设λ是一个不超过n个部分的划分。设β是一个弱递增的n元组，其元素来自{1，…n}。标记的Schur函数在变量x1，…，以λ和β为索引的xn被定义为形状为λ的半标准Young表的含量权单项式的和，其值以β的项为界。Gessel和Viennot给出了以λ和β为索引的标记Schur函数的行列式；这是可以做到的，因为这对（λ, β）满足他们的“不可置换”条件，对于n元组的某些晶格路径的末端序列，他们用来模拟tableaux。通过去掉β弱递增的要求，我们推广了标记Schur函数的概念。在此基础上，给出了λ和β的项为不可置换对（λ, β）的充分必要条件。当λ的各部分不相同时，将会有多个行界n元组通过λ上的表权构造和产生相同的多项式。据此，我们将边界n元组划分为等价类，并在每个等价类中找出最有效的n元组进行行列式计算。我们最近展示了许多其他以n和λ为索引的对象集合是通过这些有效的n元组的数量来枚举的。值得注意的是，由312-避免排列索引的GL(n) demmazure特征（关键多项式）也可以用这些行列式表示。
Let λ be a partition with no more than n parts. Let β be a weakly increasing n-tuple with entries from {1, ..., n}. The flagged Schur function in the variables x1, ..., xn that is indexed by λ and β has been defined to be the sum of the content weight monomials for the semistandard Young tableaux of shape λ whose values are rowwise bounded by the entries of β. Gessel and Viennot gave a determinant expression for the flagged Schur function indexed by λ and β; this could be done since the pair (λ, β) satisfied their “nonpermutable” condition for the sequence of terminals of an n-tuple of certain lattice paths that they used to model the tableaux. We generalize the notion of flagged Schur function by dropping the requirement that β be weakly increasing. Then we give a condition on the entries of λ and β for the pair (λ, β) to be nonpermutable that is both necessary and sufficient. When the parts of λ are not distinct there will be multiple row bound n-tuples that will produce the same polynomial via the sum of tableau weights construction on λ. We accordingly group the bounding n-tuples into equivalence classes and identify the most efficient n-tuple in each class for the determinant computation. We have recently shown that many other sets of objects that are indexed by n and λ are enumerated by the number of these efficient n-tuples. It is noted that the GL(n) Demazure characters (key polynomials) indexed by 312-avoiding permutations can also be expressed with these determinants.