Exercise 9, § 5

Tag 03ML, permanent. Cite it as [Bourbaki, Tag 03ML].

Theory of Sets, III, 237

(a) Let E be a set with $qn$ elements. Show that the number of partitions of E into $n$ subsets each of $q$ elements is equal to

(qn)!/(n!(q!)^n).

(b) Suppose that $\mathrm{E} = [1,\, qn]$ . Show that the number of partitions of E into $n$ subsets of $q$ elements, no one of which is an interval, is equal to

\frac{(qn)!}{n!(q!)^n} - \frac{(qn-q+1)!}{1!(n-1)!(q!)^{n-1}} + \frac{(qn-2q+2)!}{2!(n-2)!(q!)^{n-2}} - \cdots + (-1)^n

(same method as in Exercises 7 and 8).

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