仿射cluster代数的完美匹配与相容对之间的对应关系研究
On the correspondence between perfect matchings and compatible pairs for affine cluster algebra
AI总结：
本研究针对仿射型$A_1^{(1)}$的cluster代数，通过计数蛇形图完美匹配与极大戴克路径相容对，发现两类组合结构对应奇数下标斐波那契数，进而建立三种组合模型间的显式双射对应。
摘要：
我们采用两种方法研究仿射型$A_1^{(1)}$的cluster代数，分别是计数蛇形图（snake graph）上的完美匹配数量，以及极大戴克路径（maximal Dyck path）上的相容对数量。研究发现，这些cluster变量的洛朗多项式中各项的系数之和为奇数下标的斐波那契数。此外，我们证明了偶数长度的非递减戴克路径的数量同样是奇数下标的斐波那契数。基于此，我们明确构建了三种组合模型之间的双射对应关系，分别为蛇形图上的完美匹配、极大戴克路径上的相容对，以及偶数长度的非递减戴克路径。
英文摘要：
We study cluster algebra of affine type $A_1^{(1)}$ by using two methods including counting the numbers of perfect matchings on snake graphs and compatible pairs on maximal Dyck paths. We find that the sum of coefficients of the terms in the Laurent polynomials of these cluster variables are odd-indexed Fibonacci numbers. In addition, we prove that the numbers of non-decreasing Dyck paths of even lengths are also odd-indexed Fibonacci numbers. As a consequence, we define explicit bijective correspondences among three combinatorial models, including perfect matchings on the snake graph, compatible pairs on the maximal Dyck path, and non-decreasing Dyck paths of even lengths.
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