Successions in integer partitions

A partition of an integer n is a representation n=a (1)+a (2)+a <...a <...a <...+a (k) , with integer parts 1a parts per thousand currency signa (1)a parts per thousand currency signa (2)a parts per thousand currency signaEuro broken vertical bar a parts per thousand currency signa (k) . For any fixed positive integer p, a p-succession in a partition is defined to be a pair of adjacent parts such that a (i+1)-a (i) =p. We find generating functions for the number of partitions of n with no p-successions, as well as for the total number of such successions taken over all partitions of n. In the process, various interesting partition identities are derived. In addition, the Hardy-Ramanujan asymptotic formula for the number of partitions is used to obtain an asymptotic estimate for the average number of p-successions in the partitions of n.