{ "data": { "question": { "questionId": "1725", "questionFrontendId": "1621", "categoryTitle": "Algorithms", "boundTopicId": 448363, "title": "Number of Sets of K Non-Overlapping Line Segments", "titleSlug": "number-of-sets-of-k-non-overlapping-line-segments", "content": "

Given n points on a 1-D plane, where the ith point (from 0 to n-1) is at x = i, find the number of ways we can draw exactly k non-overlapping line segments such that each segment covers two or more points. The endpoints of each segment must have integral coordinates. The k line segments do not have to cover all n points, and they are allowed to share endpoints.

\n\n

Return the number of ways we can draw k non-overlapping line segments. Since this number can be huge, return it modulo 109 + 7.

\n\n

 

\n

Example 1:

\n\"\"\n
\nInput: n = 4, k = 2\nOutput: 5\nExplanation: The two line segments are shown in red and blue.\nThe image above shows the 5 different ways {(0,2),(2,3)}, {(0,1),(1,3)}, {(0,1),(2,3)}, {(1,2),(2,3)}, {(0,1),(1,2)}.\n
\n\n

Example 2:

\n\n
\nInput: n = 3, k = 1\nOutput: 3\nExplanation: The 3 ways are {(0,1)}, {(0,2)}, {(1,2)}.\n
\n\n

Example 3:

\n\n
\nInput: n = 30, k = 7\nOutput: 796297179\nExplanation: The total number of possible ways to draw 7 line segments is 3796297200. Taking this number modulo 109 + 7 gives us 796297179.\n
\n\n

 

\n

Constraints:

\n\n\n", "translatedTitle": "大小为 K 的不重叠线段的数目", "translatedContent": "

给你一维空间的 n 个点，其中第 i 个点（编号从 0 到 n-1）位于 x = i 处，请你找到 恰好 k 个不重叠 线段且每个线段至少覆盖两个点的方案数。线段的两个端点必须都是 整数坐标 。这 k 个线段不需要全部覆盖全部 n 个点，且它们的端点 可以 重合。

\n\n

请你返回 k 个不重叠线段的方案数。由于答案可能很大，请将结果对 109 + 7 取余 后返回。

\n\n

 

\n\n

示例 1：

\n\"\"\n
\n输入：n = 4, k = 2\n输出：5\n解释：\n如图所示，两个线段分别用红色和蓝色标出。\n上图展示了 5 种不同的方案 {(0,2),(2,3)}，{(0,1),(1,3)}，{(0,1),(2,3)}，{(1,2),(2,3)}，{(0,1),(1,2)} 。
\n\n

示例 2：

\n\n
\n输入：n = 3, k = 1\n输出：3\n解释：总共有 3 种不同的方案 {(0,1)}, {(0,2)}, {(1,2)} 。\n
\n\n

示例 3：

\n\n
\n输入：n = 30, k = 7\n输出：796297179\n解释：画 7 条线段的总方案数为 3796297200 种。将这个数对 109 + 7 取余得到 796297179 。\n
\n\n

示例 4：

\n\n
\n输入：n = 5, k = 3\n输出：7\n
\n\n

示例 5：

\n\n
\n输入：n = 3, k = 2\n输出：1
\n\n

 

\n\n

提示：

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