Record statistics in a random composition

A compositionσ=a1a2…am of n is an ordered collection of positive integers whose sum is n. An element ai in σ is a strong (weak) record if ai>aj (ai≥aj) for all j=1,2,…,i−1. Furthermore, the position of this record is i. We derive generating functions for the total number of strong (weak) records in all compositions of n, as well as for the sum of the positions of the records in all compositions of n, where the parts ai belong to A=[d]:={1,2,…,d} or A=N. In particular when A=N, we find the asymptotic mean values for the number, and for the sum of positions of records in compositions of n.