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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.

Question

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).

Expert Solution

Step 1

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

$p {(n)}^{2} \leq p (n^{2} + 2 n)$

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