1866. Number of Ways to Rearrange Sticks With K Sticks Visible

Problem

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* 10^9 + 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 10^9 + 7) ways to rearrange the sticks such that exactly 11 sticks are visible.

  Constraints:

• 1 <= n <= 1000

• 1 <= k <= n

Solution

class Solution {
    private static final int MOD = 1_000_000_007;

    public int rearrangeSticks(int n, int k) {
        if (k > n || k < 1) {
            return 0;
        }
        if (k == n) {
            return 1;
        }
        long[] dp = new long[k + 1];
        Arrays.fill(dp, 1);
        for (int i = 1; i + k <= n; i++) {
            long[] dp2 = new long[k + 1];
            for (int j = 1; j <= k; j++) {
                dp2[j] = (dp2[j - 1] + (i + j - 1) * dp[j]) % MOD;
            }
            dp = dp2;
        }
        return (int) dp[k];
    }
}

Explain:

nope.

Complexity:

• Time complexity : O(n).
• Space complexity : O(n).