Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Euler and beyond
- 3 Ferrers graphs
- 4 The Rogers-Ramanujan identities
- 5 Generating functions
- 6 Formulas for partition functions
- 7 Gaussian polynomials
- 8 Durfee squares
- 9 Euler refined
- 10 Plane partitions
- 11 Growing Ferrers boards
- 12 Musings
- A On the convergence of infinite series and products
- B References
- C Solutions and hints to selected exercises
- Index
C - Solutions and hints to selected exercises
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Euler and beyond
- 3 Ferrers graphs
- 4 The Rogers-Ramanujan identities
- 5 Generating functions
- 6 Formulas for partition functions
- 7 Gaussian polynomials
- 8 Durfee squares
- 9 Euler refined
- 10 Plane partitions
- 11 Growing Ferrers boards
- 12 Musings
- A On the convergence of infinite series and products
- B References
- C Solutions and hints to selected exercises
- Index
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.
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- Chapter
- Information
- Integer Partitions , pp. 132 - 138Publisher: Cambridge University PressPrint publication year: 2004