Problem p03820 - Generation 1

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 ≤ 2000), representing some quantity or size.
An integer K (1 ≤ K ≤ N), representing some specific position.

### 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's a Python function to generate test cases for the given problem. This function will create random values of \(N\) and \(K\) within the constraints specified, and return them as a tuple.

```python
import random

def tcgen():
    N = random.randint(1, 2000)
    K = random.randint(1, N)
    
    return N, K

# Example usage
for _ in range(5):
    print(tcgen())
```

This function generates random values for \(N\) and \(K\) within the given constraints (1 ≤ \(K\) ≤ \(N\) ≤ 2000). The `random.randint` function ensures that the values are uniformly distributed over their respective ranges. You can run the `tcgen` function multiple times to generate different test cases.

Here is a brief explanation of the function:

- `N = random.randint(1, 2000)`: This line generates a random integer \(N\) such that 1 ≤ \(N\) ≤ 2000.
- `K = random.randint(1, N)`: This line generates a random integer \(K\) such that 1 ≤ \(K\) ≤ \(N\).

Each call to `tcgen()` will produce a tuple \((N, K)\) which represents a test case for the problem.

You can use this test case generator to create multiple test cases for the problem and test your solution against them.

Extract Arguments

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