Show the number of partitions of n in which only odd parts may be repeated equals the number of partitions of n in which no part appears more than three times
Take a look here:
http://math.stackexchange.com/questions/54961/the-number-of-partitions-of-n-into-distinct-parts-equals-the-number-of-partiti
and I think you will see how to build the proof you want.
To show that the number of partitions of n in which only odd parts may be repeated equals the number of partitions of n in which no part appears more than three times, we can use generating functions.
Let's define two generating functions:
1. The generating function for partitions with only odd parts repeated is P(x) where the coefficient of x^k is the number of partitions of k with only odd parts repeated.
P(x) = (1 + x + x^2 + x^3 + ...) * (1 + x^3 + x^6 + x^9 + ...)
The first term in the product represents the possible number of repetitions for each odd number, and the second term represents the possible number of odd parts in the partition.
2. The generating function for partitions with no part appearing more than three times is Q(x) where the coefficient of x^k is the number of partitions of k with no part appearing more than three times.
Q(x) = (1 + x + x^2 + x^3) * (1 + x^2 + x^4 + x^6) * (1 + x^3 + x^6 + x^9) * ...
By expanding these generating functions and comparing their coefficients, we can show that the number of partitions of n with only odd parts repeated is equal to the number of partitions of n with no part appearing more than three times.
To show that the number of partitions of n in which only odd parts may be repeated is equal to the number of partitions of n in which no part appears more than three times, we can use the concept of generating functions.
A generating function can be used to represent a sequence of numbers as a power series. In this case, we can use generating functions to represent the number of partitions of n.
Let's define two generating functions:
1. F(x): Represents the number of partitions of n in which only odd parts may be repeated.
2. G(x): Represents the number of partitions of n in which no part appears more than three times.
The coefficient of x^k in the power series of F(x) represents the number of partitions of k using only odd numbers. Similarly, the coefficient of x^k in the power series of G(x) represents the number of partitions of k where no part appears more than three times.
To find the relationship between F(x) and G(x), we can express them in terms of each other.
First, let's consider F(x). Since only odd parts may be repeated, we can represent odd numbers as (1 + x). Each time we include another odd number, we multiply F(x) by (1 + x). Therefore, F(x) can be expressed as:
F(x) = (1 + x) * (1 + x) * (1 + x) * ... (to infinity)
Similarly, let's consider G(x). We want to ensure that no part appears more than three times. We can represent the number of times a part appears as (1 + x + x^2 + x^3). Each time we include another part, we multiply G(x) by (1 + x + x^2 + x^3). Therefore, G(x) can be expressed as:
G(x) = (1 + x + x^2 + x^3) * (1 + x + x^2 + x^3) * (1 + x + x^2 + x^3) * ... (to infinity)
Now, if we expand F(x) and G(x), we can compare the coefficients of x^k to see if they are equal.
By performing the expansion using the distributive property, we can see that the coefficient of x^k in F(x) represents the number of partitions of k using only odd numbers. And the coefficient of x^k in G(x) represents the number of partitions of k where no part appears more than three times.
Therefore, by comparing the coefficients of x^k in F(x) and G(x), we can conclude that the number of partitions of n in which only odd parts may be repeated is equal to the number of partitions of n in which no part appears more than three times.