Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices
Trees, Forests, and Total Positivity: I. $q$-Trees and $q$-Forests Matrices
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DOI：
10.37236/10465
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发表时间：
2021-06
期刊：
影响因子：
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通讯作者：
Tomack Gilmore
中科院分区：
文献类型：
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作者：
Tomack Gilmore
我们考虑的矩阵的元素是$q$中的多项式，这些多项式是由$q$产生的，这是两个众所周知的公式的推广：$n$顶点上的$k$分量的森林；在$n+1$个顶点上有根的标记树其中$k$个子结点的个数比根结点少。我们给出了森林和树木的相应统计量的组合解释，并通过各种平面网络的构造和Lindström-Gessel-Viennot引理证明了这些矩阵在系数上是完全正的。我们也展示了这些矩阵的项到八个不定式多项式的推广，并提出了一些关于它们的行生成多项式的系数方向的汉克尔全正性的猜想。
We consider matrices with entries that are polynomials in $q$ arising from natural $q$-generalisations of two well-known formulas that count: forests on $n$ vertices with $k$ components; and rooted labelled trees on $n+1$ vertices where $k$ children of the root are lower-numbered than the root. We give a combinatorial interpretation of the corresponding statistic on forests and trees and show, via the construction of various planar networks and the Lindström-Gessel-Viennot lemma, that these matrices are coefficientwise totally positive. We also exhibit generalisations of the entries of these matrices to polynomials in eight indeterminates, and present some conjectures concerning the coefficientwise Hankel-total positivity of their row-generating polynomials.