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Problem 5: Let pr(n) be the number of integer partitions of n into k parts. Show that Pr(n) > k! (k Deduce that the total number p(n) of partitions of n grows faster (as n → o) than any polynomial function of n. (Hint: It is enough to show that, for any fixed k, we have p(n) > n* for sufficiently large n.) Problem 6: Let n be a positive integer. Prove that we have the inequality of partition numbers p(n)² < p(n² + 2n).

Problem 5: Let pr(n) be the number of integer partitions of n into k parts. Show that Pr(n) > k! (k Deduce that the total number p(n) of partitions of n grows faster (as n → o) than any polynomial function of n. (Hint: It is enough to show that, for any fixed k, we have p(n) > n* for sufficiently large n.) Problem 6: Let n be a positive integer. Prove that we have the inequality of partition numbers p(n)² < p(n² + 2n).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Problem 5: Let pr(n) be the number of integer partitions of n into k parts. Show that 1 Pr(n) > 1 k! (k – 1): Deduce that the total number p(n) of partitions of n grows faster (as n → ∞) than any polynomial function of n. (Hint: It is enough to show that, for any fixed k, we have p(n) > nk for sufficiently large n.) Problem 6: Let n be a positive integer. Prove that we have the inequality of partition numbers p(n)? < p(n? + 2n).
Transcribed Image Text:Problem 5: Let pr(n) be the number of integer partitions of n into k parts. Show that 1 Pr(n) > 1 k! (k – 1): Deduce that the total number p(n) of partitions of n grows faster (as n → ∞) than any polynomial function of n. (Hint: It is enough to show that, for any fixed k, we have p(n) > nk for sufficiently large n.) Problem 6: Let n be a positive integer. Prove that we have the inequality of partition numbers p(n)? < p(n? + 2n).
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As per bartleby guidelines, for more than one question asked, only one is to be answered, kindly resubmit the other separately.

Problem 6

Let n be a positive integer.

To prove that we have the inequality of partition numbers

p(n)2≤p(n2+2n)

 

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