论文标题
晶格路径和负重依赖重量二项式系数
Lattice paths and negatively indexed weight-dependent binomial coefficients
论文作者
论文摘要
1992年，勒布（Loeb）认为二项式系数的自然扩展为负条目，并就杂种集进行了组合解释。他表明，在这种扩展的环境中，二项式系数的许多基本属性继续存在。最近，Formichella和Straub表明，这些结果可以扩展到具有任意整数值的$ Q $ binmial系数，并通过检查$ q $ binmial系数的算术性能，进一步扩展了Loeb的工作。在本文中，我们通过晶格路径给出了另一种组合解释，并考虑了第二作者首先定义的更一般的权重依赖性二项式系数的扩展，以扩展到任意整数值。值得注意的是，Loeb，Formichella和Straub的许多结果在一般的加权环境中继续保持。我们还研究了重量依赖性二项式系数的重要特殊情况，包括普通，$ Q $和椭圆二项式系数以及基础和完全均质的对称函数。
In 1992, Loeb considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Formichella and Straub showed that these results can be extended to the $q$-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining arithmetic properties of the $q$-binomial coefficients. In this paper, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by the second author, to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. We also examine important special cases of the weight-dependent binomial coefficients, including ordinary, $q$- and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions.
