Diagonal Solutions to the 2-Toda Hierarchy

Using a combinatorial description of the Bernstein operator and its action on Schur functions, we describe the formal power series solutions to a family of partial differential equations known as the 2-Toda hierarchy. We also characterize diagonal solutions and use this to prove that a special family of formal power series, called content-type series, are solutions to the 2-Toda hierarchy. As examples, we prove that various generating series for permutation factorization problems (including the double Hurwitz problem), correlators for the Schur measure on partitions and the formal character expansion of the HCIZ integral are all solutions to the 2-Toda hierarchy.