Generalized Rank Functions and an Entropy Argument

A rank function is a function f: 2[d]→N such that f(∅)=0 and f(A)⩽ f(A∪x)⩽f(A)+1 for all A⊆[d], x∈[d]\A. Athanasiadis conjectured an upper bound on the number of rank functions on 2[d]. We prove this conjecture and generalize it to functions with bounded jumps.