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By Flajolet P., Sedgewick R.

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The answer is provided by Stirling’s formula, an important approximation originally due to James Stirling (Invitation, p. 4): 1 n n√ (n → +∞). 2π n 1 + O (29) n! = e n (Several proofs are given in this book, based on the method of Laplace, p. 759, Mellin transforms, p. 766, singularity analysis, p. ) The ratios of the exact values to Stirling’s approximations n: n! 000083 show an excellent quality of the asymptotic estimate: the error is only 8% for n = 1, less than 1% for n = 10, and less than 1 per thousand for any n greater than 100.

7. Partitions with a fixed number of parts. Let P (≤k) be the class of integer partitions with at most k summands. With our notation for restricted constructions (p. 29), this class is specified as P (≤k) = MS ET≤k (I).

Let C and P denote the class of all compositions and all partitions, respecively. Since a set can always be presented in sorted order, the difference between compositions and partitions lies in the fact that the order of summands does or does not matter. This is reflected by the use of a sequence construction (for C) against a multiset construction (for P). From this perspective, it proves convenient to regard 0 as obtained by the empty sequence of summands (k = 0), and we shall do so from now on.

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