Pushes in restricted growth functions

We define the statistic of a push for restricted growth functions (rgfs) with k blocks and use this to obtain a generating function measuring the degree to which an arbitrary rgf deviates from sorted order. Several subsidiary concepts are investigated. These are the number of already sorted columns and the concept of frictionless pushes. Using these concepts, we develop expressions in terms of Bell numbers for the total number of pushes in rgfs with n parts in both the frictionless and general case.