The Distribution of Ascents of Size d or More in Partitions of n

A partition of a positive integer n is a finite sequence of positive integers a1, a2, . . . , ak such that a1 + a2 + · · · + ak = n and ai+1 ≥ ai for all i. We say n is the size of the partition, ai is the ith part of the partition and we call p(n) the number of partitions of n. For instance the 11 partitions of n = 6 are 6, 15, 24, 33, 222, 123, 114, 1113, 1122, 11112 and 111111, i.e., p(6) = 11. We define an ascent of size d or more whenever ai+1 ≥ ai + d. In this paper we aim to look at the distribution of the number of ascents of size d or more in the partitions of n. The case for d = 0, equivalent to the number of parts in partitions of n, was first studied by P. Erdős and J. Lehner in [5]. Henceforth we will restrict our attention to the case where d ≥ 1.