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  • Print publication year: 2004
  • Online publication date: June 2012

C - Solutions and hints to selected exercises

Summary

3 An obvious bijection proving the equality p(n | even parts) = p(n/2): For any partition of n into even parts, replace every part with a part of half the size. An obvious bijection proving the equality p(n/2) = p(n | even number of each part): For any partition of n/2, replace every part by two parts of the same size.

4 Every step in the splitting/merging procedure changes the number of odd parts by an even number (+2 if an even part is split into two odd parts, -2 if two odd parts are merged, and 0 otherwise). Hence, the parity (odd or even) of the number of odd parts is the same through the entire procedure.

7 Let M be the set of all positive integers that are either a power of two or three times a power of two. Then Theorem 1 says that p(n | distinct parts in M) equals p(n | parts in {1, 3}). Obviously there are └n3┘ + 1 ways of choosing the number of 3:s in such a partition, and then there is a unique way of completing the partition with l:s.

8 If n is the smallest integer that lies in one set, say M, but not in the other, say M′, then p(n | distinct parts in M) = 1 + p(n | distinct parts in M′), for the partitions counted are identical except for the partition consisting of the part n only.