The maximum number of perfect matchings in graphs with a given degree   sequence
  We show that the number of perfect matching in a simple graph $G$ with an even number of vertices and degree sequence $d_1,d_2, ..., d_n$ is at most $\prod_{i=1}^n (d_i !)^{\frac{1}{2d_i}}$. This bound is sharp if and only if $G$ is a union of complete balanced bipartite graphs. 
Noga Alon and Shmuel Friedland