The pentagram map, introduced by R. Schwartz, is defined by the following construction: given a polygon as input, draw all of its "shortest" diagonals, and output the smaller polygon which they cut out. I will explain how the machinery of cluster algebras can be used to obtain explicit formulas for the iterates of the pentagram map. The formulas are written in terms of certain cross ratios, and involve generating functions associated with a family of posets which arose in the work of N. Elkies, G. Kuperberg, M. Larsen, and J. Propp on alternating sign matrices.