The Lacking Polynomial of the Complete Bipartite Graph: Characterization, Formulae, and Log-Concavity

This note characterizes the stochastically recurrent states of the stochastic sandpile model on complete bipartite graphs K2,n and Km,2, derives explicit formulae for their lacking polynomials, proves the log-concavity of these polynomials' coefficients, and conjectures this property for the general case.