On the correspondence between perfect matchings and compatible pairs for affine cluster algebra

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.