Alphabetic points in restricted growth functions

We define an alphabetic point in a restricted growth function as a point in the word which is larger than all preceding points and smaller than all following ones. We find the trivariate generating function for restricted growth functions which tracks the number of alphabetic points as the main statistic. The generating function is then used to find the total number of alphabetic points and thereafter, a formula for the number of restricted growth functions having only one alphabetic point (i.e., those with only the initial point 1 being alphabetic). Finally, the asymptotics for those restricted growth functions having exactly m alphabetic points as well as those having only one alphabetic point are found as the size of the word n→∞.