Demystifying a divisibility property of the Kostant partition function

We study a family of identities regarding a divisibility property of the Kostant partition function which first appeared in a paper of Baldoni and Vergne. To prove the identities, Baldoni and Vergne used techniques of residues and called the resulting divisibility property "mysterious." We prove these identities entirely combinatorially and provide a natural explanation of why the divisibility occurs. We also point out several ways to generalize the identities.