In this talk, we present several perfect matching (dimer) bijections and show their applications. First, we introduce a large class of symmetric graphs and show that the numbers of their perfect matchings are given by squares of integers or two times of them (we call such numbers “squarish”). Secondly, we answer a question posed by Corteel, Huang, and Krattenthaler on finding an explicit bijection between domino tilings of two similar regions. It turns out that the bijection is valid in more general situations, and we then present a related bijection between perfect matchings and a certain class of spanning forests. Lastly, we use another dimer bijection, previously used in Ciucu’s previous work, and deduce from a refinement of it new results on spanning trees. This is a joint work with Mihai Ciucu.
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