Stacked lattice boxes

We study the number of solutions of the Diophantine equationn=x₁x₂+x₂x₃+x₃x₄+...+x_kx_k+1 The combinatorial interpretation of this equation provides the name stacked lattices boxes. The study of these objects unites three separate threads in number theory: (1) the Liouville methods, (2) MacMahon's partitions withk different parts, (3) the asymptotics of divisor sums begun by Ingham.Partially supported by National Science Foundation Grant DMS-9206993, USA.