A smooth partition of the integer n is a set of positive integers a1,a2,…ak such that
1. k is a positive integer,
2. a1≤a2≤⋯≤ak,
3. ∑ki=1ai=n, and
4. ak−a1≤1.
Determine how many smooth partitions there are of the integer 250.
250
To determine the number of smooth partitions of the integer 250, we can use the concept of generating functions.
Step 1: Create a generating function.
Let's define a generating function F(x) for the smooth partitions of 250. The generating function represents the possibilities for each term in the partition.
Step 2: Find the equation for our generating function.
Since each term in the smooth partition can be any positive integer, the generating function will include factors for each possible term. We can start with the term a1 and continue up to ak:
F(x) = (x^1 + x^2 + x^3 + ...)(x^1 + x^2 + x^3 + ...)(x^1 + x^2 + x^3 + ...) ... (k times)
Step 3: Simplify the generating function.
Since ak - a1 ≤ 1, the highest possible term in the partition should not exceed 250. Therefore, we can rewrite the generating function as:
F(x) = (x^1 + x^2 + x^3 + ... + x^250)(x^1 + x^2 + x^3 + ... + x^249)(x^1 + x^2 + x^3 + ... + x^248) ... (k times)
Step 4: Evaluate the generating function.
The generating function represents the number of smooth partitions for each value of x. To find the number of smooth partitions of 250, we need to find the coefficient of x^250 in the simplified generating function.
Step 5: Apply techniques to find the coefficient.
To find the coefficient of x^250, we can use techniques like expanding the generating function, using the binomial theorem, or polynomial multiplication. Since this process can be computationally intensive, it is best to use a computer algebra system or programming language to assist with the calculations.
Using the techniques mentioned above, we can find the number of smooth partitions of 250.