Integer Partitions by George Andrews, Kimmo Eriksson

Chapter 2

  1. [2.2] Partitioning Identity Defintion and Example: \(p(n \mid \text{even parts})\) \(= p(n \mid \text{even number of each part})\)
  2. [2.2] Exercise 3: Show that \(p(n \mid \text{even parts})\) \(= p(n/2)\) \(= p(n \mid \text{even number of each part})\)
  3. [2.3] Euler's Identity: \(p(n \mid \text{odd parts }) = p(n \mid \text{ distinct parts})\)
  4. [2.3] Exercise 4: Show that \(p(n \mid \text{even number of odd parts})\) \(= p(n \mid \text{distinct parts, number of odd parts is even})\)
  5. [2.3] Exercise 6: Show that \(p(n \mid \text{no parts divisible by \(k\)}\) \(= p(n \mid \text{less than \(k\) copies of each part})\)
  6. [2.4] \(p(n \mid \text{parts in }\{1\}) =\) \(p(n \mid \text{distinct parts in }\{1,2,4,8,...\})\)
  7. [2.4] Theorem 1: \(p(n \mid \text{parts in }\{N\}) =\) \(p(n \mid \text{distinct parts in }\{M\})\) for \(n \geq 1\), where \(N\) is any set of integers such that no elements of \(N\) is a power of two times an element of \(N\), and \(M\) is the set containing all elements of \(N\) together with all their multiplies of powers of two.
  8. [2.4] Exercise 7: Let \(\lfloor x \rfloor\) denote the largest integer smaller than or equal to \(x\). Use Theorem 1 to prove that \(\lfloor n/3 \rfloor + 1\) is the number of partitions of \(n\) into distinct parts where each part is either a power of two or three times a power of two. [TODO]
  9. [2.5] Exercise 8: For a given set \(N\), there can be at most one set \(M\) such that \(N,M\) is an Euler pair. Why? Think backward: "If there were two different such sets, \(M\) and \(M'\), then there would have to be some smallest integer. [TODO]
  10. [2.5] Exercise 9: For a given set \(N\) where some element is a power of two times some other element say \(2^ka\) and \(a\), there can exist no set \(M\) such that \(N,M\) is an Euler pair. Why? Let \(2^ka\) be the smallest element of the above mentioned type, and show that \(M\) can be uniquely constructed so that it works successfully for all \(n > 2^ka\) but that the construction will fail for \(n = 2^k\), just as in the above example. [TODO]
  11. [2.5] Exercise 10: Show that Euler pairs can be characterized more succinctly as pairs \(N,M\) such that \(2M \subset M\) and \(N = M - 2M\). [TODO]
  12. [2.5] Exercise 11: (Andrews, 1969a) Show that the number of parititions of \(n\) into \(k\)th powers \((k < 1)\) in which no part appears more than \(k - 1\) times is always equal

Chapter 3

Other Notes

  1. Every positive integer \(n\) has a unique expression of the form \(n = 2^rm\), \(r \geq 0\), \(m\) a positive odd integer.