On number of partitions of an integer into a fixed number of positive integers

•Comparison of previous upper bounds on number of partitions of n into exactly k parts.•A new upper bound for number of partitions of n into exactly k parts.•Comparison of new upper bound with previous bounds.•A new upper bound for number of partitions of n into at most k parts.

TextThis paper focuses on the number of partitions of a positive integer n into k positive summands, where k is an integer between 1 and n. Recently some upper bounds were reported for this number in [Merca14]. Here, it is shown that these bounds are not as tight as an earlier upper bound proved in [Andrews76-1] for k≤0.42nk≤0.42n. A new upper bound for the number of partitions of n into k summands is given, and shown to be tighter than the upper bound in [Merca14] when k   is between O(nln⁡n) and n−O(nln⁡n). It is further shown that the new upper bound is also tighter than two other upper bounds previously reported in [Andrews76-1] and [Colman82]. A generalization of this upper bound to number of partitions of n into at most k summands is also presented.VideoFor a video summary of this paper, please visit http://youtu.be/Pb6lKB3MnME.