Combinatorics of Horn hypergeometric series

In the 1940s, Erdélyi discovered, but did not explain, sporadic solutions to the bivariate hypergeometric systems pioneered by Horn, Appell, and Mellin from the 1880s through the 1920s. In recent work with Alicia Dickenstein and Laura Matusevich on the general multivariate case, we describe and count all of the extra solutions combinatorially in terms of lattice points.