反不变子空间枚举和 q-Hermite Catalan 矩阵条目的 Touchard 公式

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics Pub Date : 2023-12-20 DOI:10.1016/j.aam.2023.102654

Amritanshu Prasad , Samrith Ram

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引用次数: 0

摘要

我们用有限向量空间上线性算子的不变子空间数来表示其反不变子空间数。当算子可对角化且具有不同特征值时，我们的公式给出了 q-Hermite Catalan 矩阵项的有限场解释。我们还为这些项获得了一个有趣的新证明--Touchard 公式。

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Enumeration of anti-invariant subspaces and Touchard's formula for the entries of the q-Hermite Catalan matrix

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the q-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries.

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来源期刊

Advances in Applied Mathematics 数学-应用数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

85 days

期刊介绍： Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas. Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.