Matching generating function ($Z_G(\lambda )={\sum }_k{m}_k{\lambda }^k$)

The coefficient $m_k$ counts matchings in a graph with $k$ edges. For a bipartite graph with $n$ vertices on each side, $m_n$ counts perfect matchings and equals the permanent of a matrix of its bipartite adjacency matrix. Positive activity $\lambda$ makes $Z_G$ the normalizing constant of the weighted matching Markov chain.