Partition functions

Question

i'm looking at this question.

"$n$ is positive integer, $T(n)$ is the number of the partitions with odd parts of $n$ integer.

Show that $T(n) ≡ 0 \text{ (mod } 2)$ if $n \neq \dfrac{k(3k ∓ 1)}{2}.$"

i know about this question that the numbers of type $\dfrac{k(3k∓1)}{2}$ is called Pentagonal Numbers.But I don't know how to do it. May you help me please?

Here's a tutorial and reference for typesetting math on this site. — joriki, Mar 16 '20 at 13:25

score 0 · Accepted Answer · answered Mar 16 '20 at 16:33

0

The number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts. By the Pentagonal number theorem, the excess of the number of paritions of $n$ into an even number of distinct parts over those with an odd number of distinct parts is $\pm1$ if $n$ is a pentagonal number and $0$ otherwise. It follows that the parity of the number of partitions of $n$ into distinct parts (and hence into odd parts) is odd if $n$ is a pentagonal number and even otherwise.

answered Mar 16 '20 at 16:33

joriki

238,052

thank you:) i will investigate your answer. – Mar 16 '20 at 17:13
however i want to ask that this answer is enough to explain this question? we can need any teoric proof? or is there proof it in another way? – Mar 16 '20 at 17:19
@shiningstar_: I'm afraid I don't understand your question. I assume that by "teoric" you mean "theoretical" – I don't know by what criteria you judge a proof as being theoretical. What is enough as an answer depends on what you want to use it for – if you want to understand the proof, then it depends on whether you understand it. If you want to use it as an answer in an exam or homework, it depends on what's being expected. I can't envisage any scenario where I could plausibly tell you what's enough instead of you telling me. – joriki Mar 16 '20 at 17:32
I am surprised now.Because Your answer was valuable and a different perspective for me, Thanks again but i couldn't wait an answer like that. Also I think that You don't need any complicated and complex answers. You can say Yes or No. its quite simple.you know here is a knowledge and information site.so i want this question for my homework and of course i have to investigate solves in another ways.it's very normal. anyway, i have no time now, because i am very busy to search my homeworks. – Mar 16 '20 at 18:10

Partition functions

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