The main result of this paper states that in a rooted product of a path with rooted graphs which are disposed in a somewhat mirror-symmetric fashion, there are distinct eigenvalues supported in the end vertices of the path which are too close to each other: their difference is smaller than the square root of two in the even distance case, and smaller than one in the odd distance case. As a first application, we show that these end vertices cannot be involved in a quantum walk phenomenon known as perfect state transfer, significantly strengthening a recent result by two of the authors along with Godsil and van Bommel. For a second application, we show that there is no balanced integral tree of odd diameter bigger than three, answering a question raised by Híc and Nedela in 1998. Our main technique involves manipulating ratios of characteristic polynomials of graphs and subgraphs into continued fractions, and exploring in detail their analytic properties. We will also make use of a result due to Pólya and Szegö about functions that preserve the Lebesgue measure, which as far as we know is a novel application to combinatorics. In the end, we connect our machinery to a recently introduced algorithm to locate eigenvalues of trees, and with our approach we show that any graph which contains two vertices separated by a unique path that is the subdivision of a bridge with at least six inner vertices cannot be integral. As a minor corollary this implies that most trees are not integral, but we believe no one thought otherwise.
Decorated Dyck paths, polyominoes, and the Delta conjecture
我们讨论了装饰Dyck路径和装饰Dyck多角形的组合，将[Haglund 2004]和[Aval et al. 2014]的主要结果扩展到装饰的情况。这特别解决了Haglund，Remmel和Wilson（2018）的Delta猜想的情况$\langle\cdot,e_{n-d}h_d\rangle$和$\langle\cdot,h_{n-d}h_d\rangle$。在此过程中，我们引入了一些新的统计量，制定了一些新的公式，证明了对称函数的一些新恒等式，并回答了文献中的一些开放问题（例如[Haglund et al. 2018]，[Zabrocki 2016]，[Aval et al. 2015]）。主要的技术工具是麦克唐纳多项式理论中的一个新的恒等式，它扩展了Haglund在[Haglund 2004]中的一个定理。这是arXiv：1712.08787和arXiv：1709.08736的编辑合并
Decorated Dyck paths, the Delta conjecture, and a new q,t-square
我们通过引入两个新的统计量“bounce和bounce”来讨论Haglund、Remmel和Wilson（2015）和Zabrocki（2016）的Delta猜想框架中出现的装饰Dyck路径的组合学。然后，我们通过这种方式证明Haglund的q，t-Schroeder定理的修饰版本，回答Haglund，Remmel和Wilson（2015）中的一个问题，为它们的q，t-枚举数提供了体积公式。特别是，我们提供了Haglund，Remmel和Wilson（2015）和Zabrocki（2016）中所描述的对称性的代数和组合解释。最后，我们展示了这些结果与Can and Loehr（2006）的q，t-square定理（由Loehr and Warrington（2007）证明）之间的联系，得到了一个新的q，t-square，其在t=1/q的特化与经典的相同，并暗示了这种类型的更广泛的现象。
倾斜Dyck路径是Dyck路径的变体，其中除了步骤$(1,1)$和$(1,-1)$之外，还允许西南步骤$(-1,-1)$，只要路径不与自身相交。用红色的东南台阶代替西南台阶，我们最终得到了装饰过的戴克小径。Encyclopedia of Escherichia Sequences的Sequence A128723考虑了这样的路径，其中在1级的峰值是禁止的。我们提供了一个更一般的情况下，即部分装饰戴克路径，结束在规定的水平$j$，无论是从左到右，从右到左（装饰戴克路径是不对称的）的彻底分析。该方法完全基于生成函数。
College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）；Key Laboratory of Nonlinear Analysis and its Applications (Chongqing University), Ministry of Education（非线性分析及其应用教育部重点实验室（重庆大学））；Center for Combinatorics, LPMC, Nankai University（南开大学组合数学中心、理论力学与系统科学研究所）
School of Mathematics and Statistics, Beijing Institute of Technology（北京理工大学数学与统计学院）；MIIT Key Laboratory of Mathematical Theory and Computation in Information Security, Beijing Institute of Technology（北京理工大学工业和信息化部信息安全数学理论与计算重点实验室）
本文研究了联图的色对称函数，通过组合方法得到了几种图类的色对称函数展开式，并提出了帽链的 e 正猜想。
