Number of Ways to Rearrange Sticks With K Sticks Visible

Problem Description

Given n uniquely-sized sticks whose lengths are integers from 1 to n, arrange them such that exactly k sticks are visible from the left. A stick is visible from the left if there are no longer sticks to its left. Return the number of such arrangements modulo 10^9 + 7.

Key Insights

• A stick is visible if it is the tallest among all sticks to its left.
• The problem can be formulated using combinatorial mathematics.
• The number of ways to choose which k sticks are visible can be calculated using combinations.
• The remaining sticks can be arranged in any order behind the visible ones.
• Dynamic programming can be employed for efficient computation of the arrangements.

Space and Time Complexity

Time Complexity: O(n^2)
Space Complexity: O(n)

Solution

The solution involves a dynamic programming approach where we maintain a table to compute the number of valid arrangements. We define dp[i][j] as the number of ways to arrange i sticks such that exactly j are visible.

1. Initialize dp[1][1] to 1 since there's only one way to arrange one stick.
2. For each number of sticks from 2 to n, and for each possible number of visible sticks from 1 to min(k, i), update the dp table based on the previous counts.
3. The final answer will be stored in dp[n][k], which represents the arrangements of n sticks with exactly k visible.

Code Solutions