(2) Prove that for all k < n, Pk(n) < (n – k + 1)*-1. Hint: Use the previ- ous recurrence. For full credit you must clearly write the correct induction hypothesis!

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Please answer part 2 using the recurrence in the hint

Question 3. For k <n define pr(n) to be the number of integer partitions of n
with exactly k parts.
Prove the following:
(1) Prove that for all k < n, pr(n) = Pk-1(n – 1) + pr(n – k). Hint: Break
up the counting into two cases. For the first case, assume that the smallest
piece in the partition has size 1. For the second case, consider everything
not in the first case. How can you use the fact that the smallest piece has
%3D
size at least 2?
(2) Prove that for all k < n, pr (n) < (n – k + 1)k-1. Hint: Use the previ-
ous recurrence. For full credit you must clearly write the correct induction
hypothesis!
Transcribed Image Text:Question 3. For k <n define pr(n) to be the number of integer partitions of n with exactly k parts. Prove the following: (1) Prove that for all k < n, pr(n) = Pk-1(n – 1) + pr(n – k). Hint: Break up the counting into two cases. For the first case, assume that the smallest piece in the partition has size 1. For the second case, consider everything not in the first case. How can you use the fact that the smallest piece has %3D size at least 2? (2) Prove that for all k < n, pr (n) < (n – k + 1)k-1. Hint: Use the previ- ous recurrence. For full credit you must clearly write the correct induction hypothesis!
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