Enumeration of Perfect Matchings in Graphs with Reflective Symmetry

A plane graph is called symmetric if it is invariant under the reflection across some straight line. We prove a result that expresses the number of perfect matchings of a large class of symmetric graphs in terms of the product of the number of matchings of two subgraphs. When the graph is also centrally symmetric, the two subgraphs are isomorphic and we obtain a counterpart of Jockusch’s squarishness theorem. As applications of our result, we enumerate the perfect matchings of several families of graphs and we obtain new solutions for the enumeration of two of the ten symmetry classes of plane partitions (namely, transposed complementary and cyclically symmetric, transposed complementary) contained in a given box. Finally, we consider symmetry classes of perfect matchings of the Aztec diamond graph and we solve the previously open problem of enumerating the matchings that are invariant under a rotation by 90 degrees. 0. Introduction The starting point of this paper is a result [18, Theorem 1] concerning domino tilings of the Aztec diamond compatible with certain barriers. This result has also been generalized and proved bijectively by Propp [17]. We present (see Lemma 1.1) a further generalization, which allows us to prove a basic factorization theorem for the number of perfect matchings of plane bipartite graphs with a certain type of symmetry. As a direct consequence, we obtain a counterpart of Jockusch’s squarishness theorem [8, Theorem 1]. We then use the factorization theorem to enumerate the perfect matchings of several families of graphs that either generalize or are concerned with the Aztec diamond. Furthermore, we obtain new solutions for the enumeration of two of the ten symmetry classes of plane partitions contained in a given box. Motivated by the example of plane partitions, in the last section we consider the enumeration of perfect matchings of the Aztec diamond graph that are invariant under certain symmetries. There are a total of five enumerative problems that arise in this way. Two of them have been previously considered (one of which corresponds to matchings invariant Supported by a Rackham Predoctoral Fellowship at the University of Michigan l a a b b a b 1 1 2 2 3 3 Figure 1.1 Figure 1.2 under the trivial group). We present a solution for a previously open case and a new proof for the previously solved non-trivial case. 1. A Factorization Theorem A perfect matching of a graph is a collection of vertex-disjoint edges that are collectively incident to all vertices. We will often refer to a perfect matching simply as a matching. Let G be a plane graph. We say that G is symmetric if it is invariant under the reflection across some straight line. Figure 1.1 shows an example of a symmetric graph. Clearly, a symmetric graph has no perfect matching unless the axis of symmetry contains an even number of vertices (otherwise, the total number of vertices is odd); we will assume this throughout the paper. A weighted symmetric graph is a symmetric graph equipped with a weight function on the edges that is constant on the orbits of the reflection. The width of a symmetric graph G, denoted w(G), is defined to be half the number of vertices of G lying on the symmetry axis. Let G be a weighted symmetric graph with symmetry axis l, which we consider to be horizontal. Let a1, b1, a2, b2, . . . , aw(G), bw(G) be the vertices lying on l, as they occur from left to right. A reduced subgraph of G is a graph obtained from G by deleting at each vertex ai either all incident edges above l (we refer to this operation for short as “cutting above ai”) or all incident edges below l (“cutting below l,” for short). Figure 1.2 shows a reduced subgraph of the graph presented in Figure 1.1 (the deleted edges of the original graph are represented by dotted lines). The weight of a matching μ is defined to be the product of the weights of the edges contained in μ. The matching generating function of a weighted graph G, denoted M(G), is the sum of the weights of all matchings of G. The matching generating function is clearly multiplicative with respect to disjoint unions of graphs. We will henceforth assume that all graphs under consideration are connected. Lemma 1.1. All 2 reduced subgraphs of a weighted symmetric graph G have the same matching generating function. Proof. It is enough to prove the statement of the Lemma for two reduced subgraphs that differ only around a single vertex ai. Let G1 and G2 be two reduced subgraphs obtained by identical cutting operations except that for the former we made a cut above ai, while for the latter we cut below ai (for some i ∈ {1, 2, . . . , w(G)}). Let μ be a matching of G1 and let μ be the matching of G obtained from μ by reflection across l. Then ν = μ∪μ (where