A smooth partition of the integer n is a set of positive integers a 1 ,a 2 ,…a k such that
1. k is a positive integer,
2. a 1 ≤a 2 ≤⋯≤a k ,
3. ∑ k i=1 a i =n, and
4. a k −a 1 ≤1.
Determine how many smooth partitions there are of the integer 250.
Note: The definition given above is not standard terminology.
250
How? Pls explain
CHEATER!!!!!
do not answer this. this is an ongoing problem on brilliant . org.
Be ashamed of yourself, "best".
To determine the number of smooth partitions of the integer 250, we can use a dynamic programming approach.
Let's define a function P(n, m) which represents the number of smooth partitions of the integer n with the largest element being at most m.
We can calculate P(n, m) using the following recurrence relation:
P(n, m) = P(n - m, m) + P(n, m - 1)
Here's how this recurrence relation works:
- The first term, P(n - m, m), represents the number of smooth partitions of the integer n - m with the largest element being m. We subtract m from n to exclude all partitions that have m as the largest element.
- The second term, P(n, m - 1), represents the number of smooth partitions of the integer n with the largest element being less than m. We subtract 1 from m to consider partitions without m as the largest element.
To calculate P(n, m), we start with the base case P(0, m) = 1 for all m. This indicates that there is exactly one smooth partition of the integer 0, which is an empty set.
Using the recurrence relation, we can calculate P(n, m) for all values of n from 1 to 250 and m from 1 to n. Finally, the answer to the question is the value of P(250, 250).
Here's the Python code to implement this dynamic programming approach:
def smooth_partitions(n):
dp = [[0] * (n+1) for _ in range(n+1)]
# Base case: there is one smooth partition of 0 (an empty set)
for m in range(1, n+1):
dp[0][m] = 1
# Calculate P(n, m) for all values of n from 1 to 250 and m from 1 to n
for i in range(1, n+1):
for j in range(1, i+1):
dp[i][j] = dp[i-j][j] + dp[i][j-1]
return dp[n][n]
num_partitions = smooth_partitions(250)
print(num_partitions)
By running the above code, you will get the number of smooth partitions of the integer 250.