160. () (a) What is the generating function (using q for the variable) for the n

Question

160. () (a) What is the generating function (using q for the variable) for the number of partitions (b) What is the generating function (using q as the variable) for the number of partitions of (c) Use generating functions to explain why the number of partitions of an integer in which of an integer in which each part is even? an integer into distinct parts, that is, in which each part is used at most once? each part is used an even number of times equals the generating function for the number of partitions of an integer in which each part is even. Compare your answer to Problem 131.

Explanation / Answer

lets us consider that the integar is divided into n equal parts

a)the generating function all prts becomes even is nCn/2

b) the generating function each part is used at ost once is nC1+nC2+nC3+.........+nCn

c)because the number partions are n the numbe of even partions are equal to be n/2 and the number of odd partions are n/2

and the number of times used aprtion at even numbr of times is equal to n/2

then the both are equal

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