Problem p03820 - Generation 3

Orig Description

Score : 1200 points
Problem StatementSnuke has decided to play with N cards and a deque (that is, a double-ended queue).
Each card shows an integer from 1 through N, and the deque is initially empty.
Snuke will insert the cards at the beginning or the end of the deque one at a time, in order from 1 to N.
Then, he will perform the following action N times: take out the card from the beginning or the end of the deque and eat it.
Afterwards, we will construct an integer sequence by arranging the integers written on the eaten cards, in the order they are eaten. Among the sequences that can be obtained in this way, find the number of the sequences such that the K-th element is 1. Print the answer modulo 10^{9} + 7.
Constraints
1 ≦ K ≦ N ≦ 2{,}000
InputThe input is given from Standard Input in the following format:
N K
OutputPrint the answer modulo 10^{9} + 7.
Sample Input 12 1
Sample Output 11
There is one sequence satisfying the condition: 1,2. One possible way to obtain this sequence is the following:
Insert both cards, 1 and 2, at the end of the deque.
Eat the card at the beginning of the deque twice.
Sample Input 217 2
Sample Output 2262144
Sample Input 32000 1000
Sample Output 3674286644

Extracted Specification

An integer N (1 ≤ N ≤ 2,000), representing some quantity or size.
An integer K (1 ≤ K ≤ N), representing some quantity or index.

### Example Input:

```
12 1
```

### Function Signature:
Write a function f(N, K) that takes in the input.
def f(N: int, K: int): 
    ''' 
    N: an integer 
    K: an integer
    '''

Test Case Generator

Sure, here is a test case generator for the given problem statement:

```python
import random

def tcgen():
    # Constants
    MAX_N = 2000
    MODULO = 10**9 + 7
    
    # Randomly generate N and K
    N = random.randint(1, MAX_N)
    K = random.randint(1, N)
    
    return N, K

# Example usage
if __name__ == "__main__":
    for _ in range(5):  # Generate 5 test cases for demonstration
        print(tcgen())
```

### Explanation:

1. **Constants**:
   - `MAX_N = 2000`: The maximum value for \( N \), as per the problem constraints.
   - `MODULO = 10^9 + 7`: The modulo value used for the output, though it's not directly used in generating test cases, it's useful for any further calculations.

2. **Generating Random Values**:
   - `N`: Randomly generated between 1 and 2000.
   - `K`: Randomly generated between 1 and \( N \).

3. **Returning the Test Case**:
   - The function returns a tuple `(N, K)`.

4. **Example Usage**:
   - The example usage demonstrates how to generate and print 5 test cases.

This generator will produce random test cases within the specified constraints, allowing you to test the robustness and performance of your solution.

Extract Arguments

def extract_arguments(fh):
    N, K = map(int, fh.readline().strip().split())
    return N, K