There are n uniquely-sized sticks whose lengths are integers from 1 to n. You want to arrange the sticks such that exactlyk sticks are visible from the left. A stick is visible from the left if there are no longer sticks to the left of it.
For example, if the sticks are arranged [1,3,2,5,4], then the sticks with lengths 1, 3, and 5 are visible from the left.
Given n and k, return the number of such arrangements. Since the answer may be large, return it modulo109 + 7.
Example 1:
Input: n = 3, k = 2
Output: 3
Explanation: [1,3,2], [2,3,1], and [2,1,3] are the only arrangements such that exactly 2 sticks are visible.
The visible sticks are underlined.
Example 2:
Input: n = 5, k = 5
Output: 1
Explanation: [1,2,3,4,5] is the only arrangement such that all 5 sticks are visible.
The visible sticks are underlined.
Example 3:
Input: n = 20, k = 11
Output: 647427950
Explanation: There are 647427950 (mod 109 + 7) ways to rearrange the sticks such that exactly 11 sticks are visible.
Constraints:
1 <= n <= 1000
1 <= k <= n
Solutions
Solution 1: Dynamic Programming
We define $f[i][j]$ to represent the number of permutations of length $i$ in which exactly $j$ sticks can be seen. Initially, $f[0][0]=1$ and the rest $f[i][j]=0$. The answer is $f[n][k]$.
Consider whether the last stick can be seen. If it can be seen, it must be the longest. Then there are $i - 1$ sticks in front of it, and exactly $j - 1$ sticks can be seen, which is $f[i - 1][j - 1]$. If the last stick cannot be seen, it can be any one except the longest stick. Then there are $i - 1$ sticks in front of it, and exactly $j$ sticks can be seen, which is $f[i - 1][j] \times (i - 1)$.
We notice that $f[i][j]$ is only related to $f[i - 1][j - 1]$ and $f[i - 1][j]$, so we can use a one-dimensional array to optimize the space complexity.
The time complexity is $O(n \times k)$, and the space complexity is $O(k)$. Here, $n$ and $k$ are the two integers given in the problem.