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Thus, this new chain partition has the desired property: A is a subset of every element of the 1 or 2 element chain C 1 , so A is not in an anti-chain of maximum size. Finally, we need to show that if n is odd, no anti-chain of maximum size contains n n sets in both n/2 and n/2 . Suppose there is such an anti-chain, consisting of sets Ak+1 , . . , Al in in n n/2 n n/2 , where l = n n/2 , and B1 , . . , Bk in are A1 , . . , Ak , and the remaining sets in n n/2 n n/2 . The remaining sets are Bk+1 , .

Find the number of permutations of 1, 2, . . , 8 that have at least one odd number in the correct position. 6. How many permutations of [n] have exactly k numbers in their correct positions? 7. Give a combinatorial proof that n n! = k=0 n Dn−k . k 8. A small merry-go-round has 8 seats occupied by 8 children. In how many ways can the children change places so that no child sits behind the same child as on the first ride? The seats do not matter, only the relative positions of the children. 3 Generating Functions As we have seen, a typical counting problem includes one or more parameters, which of course show up in the solutions, such as nk , P (n, k), or the number of derangements of [n].

The action of conjugation takes every partition of one type into a partition of the other: the conjugate of a partition into k parts is a partition with largest part k and vice versa. This establishes a 1–1 correspondence between partitions into k parts and partitions with largest part k. 3. 1. Use generating functions to find p15 . 2. Find the generating function for the number of partitions of an integer into distinct odd parts. Find the number of such partitions of 20. 3. Find the generating function for the number of partitions of an integer into distinct even parts.

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