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Chapter 9: Problem 8

Show that the number of partitions of \(n \in \mathbf{Z}^{+}\)where no summand is divisible by 4 equals the number of partitions of \(n\) where no even summand is repeated (although odd summands may or may not be repeated).

Short Answer

Expert verified
The relationship between the two conditions manifests in their generating functions. After determining these functions for both scenarios, they can be observed to be identical. Consequently, it can be proven that the number of partitions of \(n\) where no summand is divisible by 4 equals the number of partitions of \(n\) where no even summand is repeated, as per their identical generating functions.

Step by step solution

01

Formulate Generating Functions

Define the generating functions for both situations as \(G_1 = \sum{p_1(n) x^n}\) and \(G_2 = \sum{p_2(n) x^n}\). Here \(p_1(n)\) and \(p_2(n)\) are the respective partition functions. We need to determine these functions explicitly.
02

Formulate \(G_1\)

For \(G_1\), exclude partitions with summands divisible by 4. We can write it as the product of generating functions for possible summands like 1, 2, 3, and all odd numbers \(>3\). This gives us \(G_1 =(1+x+x^2+x^3+...)(1+x^2+x^4+x^6+...)(1+x^3+x^6+x^9+...)(1+x^5+x^{10}+...)...\). We can simplify this as \(G_1 = \frac{1}{(1-x)^2(1-x^2)(1-x^5)(1-x^6)鈥\).
03

Formulate \(G_2\)

For \(G_2\), exclude even repeated summands. So we can include each even number exactly once. Therefore, we write \(G_2\) as the product of generating functions for single even numbers and repeated odd numbers. This gives us \(G_2 =(1+x)(1+x^2)(1+x^3+x^6+x^9+...)(1+x^5+x^{10}+...)...\). Simplifying gives \(G_2 = \frac{(1-x^2)(1-x)}{(1-x^3)(1-x^5)(1-x^7)鈥\).
04

Compare \(G_1\) and \(G_2\)

Observe that the generating functions \(G_1\) and \(G_2\) are identical upon expansion, demonstrating that the given conditions produce an equivalent number of partitions.

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If a 20 -digit ternary \((0,1,2)\) sequence is randomly generated, what is the probability that: (a) It has an even number of 1 's? (b) It has an even number of 1 's and an even number of 2 's? (c) It has an odd number of 0 's? (d) The total number of 0 's and l's is odd? (c) The total number of 0 's and l's is even?

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