Description#
There are n uniquely-sized sticks whose lengths are integers from 1 to n. You want to arrange the sticks such that exactly k 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 modulo 109 + 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 <= 10001 <= 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 $f[i - 1][j] \times (i - 1)$.
Therefore, the state transition equation is:
$$
f[i][j] = f[i - 1][j - 1] + f[i - 1][j] \times (i - 1)
$$
The final answer is $f[n][k]$.
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.
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| class Solution:
def rearrangeSticks(self, n: int, k: int) -> int:
mod = 10**9 + 7
f = [[0] * (k + 1) for _ in range(n + 1)]
f[0][0] = 1
for i in range(1, n + 1):
for j in range(1, k + 1):
f[i][j] = (f[i - 1][j - 1] + f[i - 1][j] * (i - 1)) % mod
return f[n][k]
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| class Solution {
public int rearrangeSticks(int n, int k) {
final int mod = (int) 1e9 + 7;
int[][] f = new int[n + 1][k + 1];
f[0][0] = 1;
for (int i = 1; i <= n; ++i) {
for (int j = 1; j <= k; ++j) {
f[i][j] = (int) ((f[i - 1][j - 1] + f[i - 1][j] * (long) (i - 1)) % mod);
}
}
return f[n][k];
}
}
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| class Solution {
public:
int rearrangeSticks(int n, int k) {
const int mod = 1e9 + 7;
int f[n + 1][k + 1];
memset(f, 0, sizeof(f));
f[0][0] = 1;
for (int i = 1; i <= n; ++i) {
for (int j = 1; j <= k; ++j) {
f[i][j] = (f[i - 1][j - 1] + (i - 1LL) * f[i - 1][j]) % mod;
}
}
return f[n][k];
}
};
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| func rearrangeSticks(n int, k int) int {
const mod = 1e9 + 7
f := make([][]int, n+1)
for i := range f {
f[i] = make([]int, k+1)
}
f[0][0] = 1
for i := 1; i <= n; i++ {
for j := 1; j <= k; j++ {
f[i][j] = (f[i-1][j-1] + (i-1)*f[i-1][j]) % mod
}
}
return f[n][k]
}
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| function rearrangeSticks(n: number, k: number): number {
const mod = 10 ** 9 + 7;
const f: number[][] = Array.from({ length: n + 1 }, () =>
Array.from({ length: k + 1 }, () => 0),
);
f[0][0] = 1;
for (let i = 1; i <= n; ++i) {
for (let j = 1; j <= k; ++j) {
f[i][j] = (f[i - 1][j - 1] + (i - 1) * f[i - 1][j]) % mod;
}
}
return f[n][k];
}
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Solution 2#
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| class Solution:
def rearrangeSticks(self, n: int, k: int) -> int:
mod = 10**9 + 7
f = [1] + [0] * k
for i in range(1, n + 1):
for j in range(k, 0, -1):
f[j] = (f[j] * (i - 1) + f[j - 1]) % mod
f[0] = 0
return f[k]
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| class Solution {
public int rearrangeSticks(int n, int k) {
final int mod = (int) 1e9 + 7;
int[] f = new int[k + 1];
f[0] = 1;
for (int i = 1; i <= n; ++i) {
for (int j = k; j > 0; --j) {
f[j] = (int) ((f[j] * (i - 1L) + f[j - 1]) % mod);
}
f[0] = 0;
}
return f[k];
}
}
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| class Solution {
public:
int rearrangeSticks(int n, int k) {
const int mod = 1e9 + 7;
int f[k + 1];
memset(f, 0, sizeof(f));
f[0] = 1;
for (int i = 1; i <= n; ++i) {
for (int j = k; j; --j) {
f[j] = (f[j - 1] + f[j] * (i - 1LL)) % mod;
}
f[0] = 0;
}
return f[k];
}
};
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| func rearrangeSticks(n int, k int) int {
const mod = 1e9 + 7
f := make([]int, k+1)
f[0] = 1
for i := 1; i <= n; i++ {
for j := k; j > 0; j-- {
f[j] = (f[j-1] + f[j]*(i-1)) % mod
}
f[0] = 0
}
return f[k]
}
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| function rearrangeSticks(n: number, k: number): number {
const mod = 10 ** 9 + 7;
const f: number[] = Array(n + 1).fill(0);
f[0] = 1;
for (let i = 1; i <= n; ++i) {
for (let j = k; j; --j) {
f[j] = (f[j] * (i - 1) + f[j - 1]) % mod;
}
f[0] = 0;
}
return f[k];
}
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