Total positivity of some polynomial matrices that enumerate labeled trees and forests II. Rooted labeled trees and partial functional digraphs
Total positivity of some polynomial matrices that enumerate labeled trees and forests II. Rooted labeled trees and partial functional digraphs
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DOI：
10.1016/j.aam.2024.102703
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发表时间：
2023-02
期刊：
影响因子：
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通讯作者：
X. Chen;A. Sokal
中科院分区：
文献类型：
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作者：
X. Chen;A. Sokal
研究了元素为tn，k=（nk）nn-k的下三角矩阵的三种组合模型：两种涉及顶点集[n+ 1]上的根树，一种涉及顶点集[n]上的部分功能有向图.我们证明了这个矩阵是完全正的，并且它的行生成多项式序列是系数Hankel-完全正的。然后，我们推广到多项式t n，k（y，z），计算不正确的和正确的边缘，并进一步推广到多项式t n，k（y，z），在无限多个不定式中，给每个不正确的边缘一个权重y和一个权重m！对于每个顶点有m个适当的子节点，我们证明了，如果权重序列是Toeplitz-完全积极的，那么上述两个完全积极的结果继续保持。我们的证明使用生产矩阵和指数Riordan阵列。
We study three combinatorial models for the lower-triangular matrix with entries t n, k=(n k) n n− k: two involving rooted trees on the vertex set [n+ 1], and one involving partial functional digraphs on the vertex set [n]. We show that this matrix is totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. We then generalize to polynomials t n, k (y, z) that count improper and proper edges, and further to polynomials t n, k (y, ϕ) in infinitely many indeterminates that give a weight y to each improper edge and a weight m! ϕ m for each vertex with m proper children. We show that if the weight sequence ϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.