The Q is: show that the number of partitions of n within Z+ where no summand is divisible by 4 equals the number of partitions of n where no even summand is repeated(adsbygoogle = window.adsbygoogle || []).push({});

Here is what I got so far

Let the partition where no summand is divisible by 4 be P1(x)

Let the partition where no even summand is repeated be P2(x)

My goal is to show that P1(x) = P2(x)

P1(x) = (1+x+x^2 ... )(1+x^2+x^4 ... )(1+x^3+x^6 ...)(1+x^5+x^10 ...)...

(skip x^4, x^8, ... etc.)

P1(x) = [ let i go from 1 to infinity, the products of ( 1 / (1-x^i) ) ] /

[ let i go from 1 to infinity, the prodcuts of ( 1 / (1-x^(4i) )

P2(x) = [ (1+x^2)(1+x^4) ... ][ (1+x+x^2 ...)(1+x^3+x^6) ... ]

P2(x) = [ let i go from 1 to infinity,the products of ( 1 + x^(2i) ) ] *

[ let i go from 1 to infinity, the products of ( 1 / (1-x^(2i-1)) )

So are my equations correct? If so, how do I solve them?

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# Homework Help: Partition of Integer need advice

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