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II. The Partition Function
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Problems for Chapter II
1. Explain the observation that MacMahon made of a parabola when
he viewed the list of the (decimal expansions) of the partition
function.
2. Prove the “simple” fact that, if order counts (e.g., 2 + 5 is consid-
ered a different partition of 7 than 5 + 2), then the total number
of partitions on n would be 2 n−1 .
3. Explain the approximation “near 1” of log 1 as 2 1−z + O (1 −
z 1+z
zð 3 . Why does this lead to
1 1 1 + z
+ O(1 − zð ?
log 1 2 1 − z
z
4. Why is the Riemann sum such a good approximation to the in-
tegral when the function is monotone and the increments are
equal?
