Generalized rank functions and an entropy argument

A rank function is a function f:2([d]) --> N such that f(O) = 0 and f(A) le ss than or equal to f(A boolean OR x) less than or equal to f(A) + 1 for al l A subset of or equal to [d], x is an element of [d]\A. Athanusiadis conje ctured an upper bound on the number of rank functions on 2([d]). We prove t his conjecture and generalize it to functions with bounded jumps. (C) 1999 Academic Press.