Estimate of the Number of Restricted Integer-partitions

ordering of the parts then this number reduces to the classical Stirling number of the second kind (see [3], p. 244). Often it is the case where there is a constraint on the size of the parts. For instance, consider ordered m-partitions of n where each part must be of size between 1 and N for some constant N ≥ 1. There are known recurrences for the number of such constrained ordered partitions (see for instance, Theorem 4.2 of [1]). It is not difficult to compute this exactly as the following lemma shows. First, we define the binomial coefficient quantity