弱增树与多集Schett多项式的对称性
A symmetry on weakly increasing trees and multiset Schett polynomials
摘要：
通过考虑增树中结点的度和水平的奇偶性，得到了雅可比椭圆函数的Taylor展开式系数的一种新的组合解释，并解决了Ma-Mansour-Wang-Yeh的一个猜想.林-马-马-周统一了增树和平面树的概念，引入了多重集上的弱增树。在弱增树上发现了“奇数层上的偶数度节点”和“奇数度节点”的对称联合分布，从而将Schett多项式推广到了多集上. Schett多项式是Schett引入的雅可比椭圆函数的推广.一个组合证明和代数证明这种对称性，以及几个相关的有趣的后果。此外，通过引入树上的群作用，证明了多集Schett多项式的部分$γ$ -正性，一个结果暗示了这些多项式的对称性和单峰性.
英文摘要：
By considering the parity of the degrees and levels of nodes in increasing trees, a new combinatorial interpretation for the coefficients of the Taylor expansions of the Jacobi elliptic functions is found. As one application of this new interpretation, a conjecture of Ma-Mansour-Wang-Yeh is solved. Unifying the concepts of increasing trees and plane trees, Lin-Ma-Ma-Zhou introduced weakly increasing trees on a multiset. A symmetry joint distribution of "even-degree nodes on odd levels" and "odd-degree nodes" on weakly increasing trees is found, extending the Schett polynomials, a generalization of the Jacobi elliptic functions introduced by Schett, to multisets. A combinatorial proof and an algebraic proof of this symmetry are provided, as well as several relevant interesting consequences. Moreover, via introducing a group action on trees, we prove the partial $γ$-positivity of the multiset Schett polynomials, a result implies both the symmetry and the unimodality of these polynomials.
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