By Jürgen Bierbrauer

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**Extra resources for Introduction to Combinatorics [Lecture notes]**

**Sample text**

De ne A to consist of the permutations containing MATH as a word, analogously A (IS is a subword) and A (FUN is a subword). The permutations in Ai are easy to count. Each element of A can be seen as a permutation of the 6 objects MATH; I; S; F; U; N: It follows jA j = 6! = 720: For the same reason we have jA j = 8! = 40; 320 and jA j = 7! = 5; 040: The sizes of the various intersections are just as easily determined. The elements of A \ A can be seen as the permutations of the 5 objects MATH; IS; F; U; N; hence jA \ A j = 5!

Set up a table of values of An; Bn in the previous problem for n 35: =0 2 =0 1 1 2 2 3 3 =0 5 5 6 4 52 CHAPTER 7. GENERATING FUNCTIONS Chapter 8 Recurrence relations We know that the number sn of subsets of an n-set is 2n: This could have been proved as well by starting from the observation that s = 1 and sn = 2sn for n > 0: This is a particularly simple example of a recurrence relation. Consider the sequence qn de ned by the recurrence relation q = 1; qn = qn + (2n + 1) for n > 1: Clearly we can describe qn as the sum of the n rst odd natural numbers.

Symmetry is clear. 5. In fact, we see mn = mn = n mm : The sequence is growing as long as this quotient is 1; equivalently n 2m 1: It is best to distinguish the cases n even and n odd. If n = 2k is even, then the largest number in the sequence +1 1 39 CHAPTER 6. 3 Corollary. The largest binomial with numerator n is bn=n2c : Here we use terminology (bxc) that was introduced in Chapter 3. Sperner's question concerns certain types of families of subsets of a given set. 4 De nition. Let F be a family of subsets of the n-set S: We call F an antichain if no two di erent elements of F are subsets of each other.