The partition function Q(n), which denotes the number of partitions of a positive integer n into distinct parts, has been the subject of a dozen papers. In this paper, we study this kind of partition with the additional constraint that the parts are bounded by a fixed integer. We denote the number of partitions of an integer n into distinct parts, each ≤ k, by Qk(n). We find a sharp upper bound for Qk(n), and more, an infinite series lower bound for the partition function Q(n). In the last section, we exhibit a group of interesting identities involving Qk(n) that arise from a combinatorial problem.
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