leetcode-problemset/leetcode-cn/originData/count-the-number-of-inversions.json

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    "data": {
        "question": {
            "questionId": "3460",
            "questionFrontendId": "3193",
            "categoryTitle": "Algorithms",
            "boundTopicId": 2817245,
            "title": "Count the Number of Inversions",
            "titleSlug": "count-the-number-of-inversions",
            "content": "<p>You are given an integer <code>n</code> and a 2D array <code>requirements</code>, where <code>requirements[i] = [end<sub>i</sub>, cnt<sub>i</sub>]</code> represents the end index and the <strong>inversion</strong> count of each requirement.</p>\n\n<p>A pair of indices <code>(i, j)</code> from an integer array <code>nums</code> is called an <strong>inversion</strong> if:</p>\n\n<ul>\n\t<li><code>i &lt; j</code> and <code>nums[i] &gt; nums[j]</code></li>\n</ul>\n\n<p>Return the number of <span data-keyword=\"permutation\">permutations</span> <code>perm</code> of <code>[0, 1, 2, ..., n - 1]</code> such that for <strong>all</strong> <code>requirements[i]</code>, <code>perm[0..end<sub>i</sub>]</code> has exactly <code>cnt<sub>i</sub></code> inversions.</p>\n\n<p>Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>\n\n<p>&nbsp;</p>\n<p><strong class=\"example\">Example 1:</strong></p>\n\n<div class=\"example-block\">\n<p><strong>Input:</strong> <span class=\"example-io\">n = 3, requirements = [[2,2],[0,0]]</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\">2</span></p>\n\n<p><strong>Explanation:</strong></p>\n\n<p>The two permutations are:</p>\n\n<ul>\n\t<li><code>[2, 0, 1]</code>\n\n\t<ul>\n\t\t<li>Prefix <code>[2, 0, 1]</code> has inversions <code>(0, 1)</code> and <code>(0, 2)</code>.</li>\n\t\t<li>Prefix <code>[2]</code> has 0 inversions.</li>\n\t</ul>\n\t</li>\n\t<li><code>[1, 2, 0]</code>\n\t<ul>\n\t\t<li>Prefix <code>[1, 2, 0]</code> has inversions <code>(0, 2)</code> and <code>(1, 2)</code>.</li>\n\t\t<li>Prefix <code>[1]</code> has 0 inversions.</li>\n\t</ul>\n\t</li>\n</ul>\n</div>\n\n<p><strong class=\"example\">Example 2:</strong></p>\n\n<div class=\"example-block\">\n<p><strong>Input:</strong> <span class=\"example-io\">n = 3, requirements = [[2,2],[1,1],[0,0]]</span></p>\n\n<p><strong>Output:</strong> 1</p>\n\n<p><strong>Explanation:</strong></p>\n\n<p>The only satisfying permutation is <code>[2, 0, 1]</code>:</p>\n\n<ul>\n\t<li>Prefix <code>[2, 0, 1]</code> has inversions <code>(0, 1)</code> and <code>(0, 2)</code>.</li>\n\t<li>Prefix <code>[2, 0]</code> has an inversion <code>(0, 1)</code>.</li>\n\t<li>Prefix <code>[2]</code> has 0 inversions.</li>\n</ul>\n</div>\n\n<p><strong class=\"example\">Example 3:</strong></p>\n\n<div class=\"example-block\">\n<p><strong>Input:</strong> <span class=\"example-io\">n = 2, requirements = [[0,0],[1,0]]</span></p>\n\n<p><strong>Output:</strong> <span class=\"example-io\">1</span></p>\n\n<p><strong>Explanation:</strong></p>\n\n<p>The only satisfying permutation is <code>[0, 1]</code>:</p>\n\n<ul>\n\t<li>Prefix <code>[0]</code> has 0 inversions.</li>\n\t<li>Prefix <code>[0, 1]</code> has an inversion <code>(0, 1)</code>.</li>\n</ul>\n</div>\n\n<p>&nbsp;</p>\n<p><strong>Constraints:</strong></p>\n\n<ul>\n\t<li><code>2 &lt;= n &lt;= 300</code></li>\n\t<li><code>1 &lt;= requirements.length &lt;= n</code></li>\n\t<li><code>requirements[i] = [end<sub>i</sub>, cnt<sub>i</sub>]</code></li>\n\t<li><code>0 &lt;= end<sub>i</sub> &lt;= n - 1</code></li>\n\t<li><code>0 &lt;= cnt<sub>i</sub> &lt;= 400</code></li>\n\t<li>The input is generated such that there is at least one <code>i</code> such that <code>end<sub>i</sub> == n - 1</code>.</li>\n\t<li>The input is generated such that all <code>end<sub>i</sub></code> are unique.</li>\n</ul>\n",
            "translatedTitle": "统计逆序对的数目",
            "translatedContent": "<p>给你一个整数&nbsp;<code>n</code>&nbsp;和一个二维数组&nbsp;<code>requirements</code>&nbsp;，其中&nbsp;<code>requirements[i] = [end<sub>i</sub>, cnt<sub>i</sub>]</code> <span class=\"text-only\" data-eleid=\"10\" style=\"white-space: pre;\">表示这个要求中的末尾下标和 <strong>逆序对</strong> 的数目。</span></p>\n\n<p>整数数组 <code>nums</code>&nbsp;中一个下标对&nbsp;<code>(i, j)</code>&nbsp;如果满足以下条件，那么它们被称为一个 <strong>逆序对</strong>&nbsp;：</p>\n\n<ul>\n\t<li><code>i &lt; j</code>&nbsp;且&nbsp;<code>nums[i] &gt; nums[j]</code></li>\n</ul>\n\n<p>请你返回&nbsp;<code>[0, 1, 2, ..., n - 1]</code>&nbsp;的&nbsp;<span data-keyword=\"permutation\">排列</span> <code>perm</code>&nbsp;的数目，满足对 <strong>所有</strong>&nbsp;的&nbsp;<code>requirements[i]</code>&nbsp;都有&nbsp;<code>perm[0..end<sub>i</sub>]</code>&nbsp;恰好有&nbsp;<code>cnt<sub>i</sub></code>&nbsp;个逆序对。</p>\n\n<p>由于答案可能会很大，将它对&nbsp;<code>10<sup>9</sup> + 7</code>&nbsp;<strong>取余</strong>&nbsp;后返回。</p>\n\n<p>&nbsp;</p>\n\n<p><strong class=\"example\">示例 1：</strong></p>\n\n<div class=\"example-block\">\n<p><span class=\"example-io\"><b>输入：</b>n = 3, requirements = [[2,2],[0,0]]</span></p>\n\n<p><span class=\"example-io\"><b>输出：</b>2</span></p>\n\n<p><strong>解释：</strong></p>\n\n<p>两个排列为：</p>\n\n<ul>\n\t<li><code>[2, 0, 1]</code>\n\n\t<ul>\n\t\t<li>前缀&nbsp;<code>[2, 0, 1]</code>&nbsp;的逆序对为&nbsp;<code>(0, 1)</code> 和&nbsp;<code>(0, 2)</code>&nbsp;。</li>\n\t\t<li>前缀&nbsp;<code>[2]</code>&nbsp;的逆序对数目为 0 个。</li>\n\t</ul>\n\t</li>\n\t<li><code>[1, 2, 0]</code>\n\t<ul>\n\t\t<li>前缀&nbsp;<code>[1, 2, 0]</code>&nbsp;的逆序对为&nbsp;<code>(0, 2)</code> 和&nbsp;<code>(1, 2)</code>&nbsp;。</li>\n\t\t<li>前缀&nbsp;<code>[1]</code>&nbsp;的逆序对数目为 0 个。</li>\n\t</ul>\n\t</li>\n</ul>\n</div>\n\n<p><strong class=\"example\">示例 2：</strong></p>\n\n<div class=\"example-block\">\n<p><span class=\"example-io\"><b>输入：</b>n = 3, requirements = [[2,2],[1,1],[0,0]]</span></p>\n\n<p><b>输出：</b>1</p>\n\n<p><strong>解释：</strong></p>\n\n<p>唯一满足要求的排列是&nbsp;<code>[2, 0, 1]</code>&nbsp;：</p>\n\n<ul>\n\t<li>前缀&nbsp;<code>[2, 0, 1]</code>&nbsp;的逆序对为&nbsp;<code>(0, 1)</code> 和&nbsp;<code>(0, 2)</code>&nbsp;。</li>\n\t<li>前缀&nbsp;<code>[2, 0]</code>&nbsp;的逆序对为&nbsp;<code>(0, 1)</code>&nbsp;。</li>\n\t<li>前缀&nbsp;<code>[2]</code>&nbsp;的逆序对数目为 0 。</li>\n</ul>\n</div>\n\n<p><strong class=\"example\">示例 3：</strong></p>\n\n<div class=\"example-block\">\n<p><span class=\"example-io\"><b>输入：</b>n = 2, requirements = [[0,0],[1,0]]</span></p>\n\n<p><span class=\"example-io\"><b>输出：</b>1</span></p>\n\n<p><b>解释：</b></p>\n\n<p>唯一满足要求的排列为&nbsp;<code>[0, 1]</code>&nbsp;：</p>\n\n<ul>\n\t<li>前缀&nbsp;<code>[0]</code>&nbsp;的逆序对数目为 0 。</li>\n\t<li>前缀&nbsp;<code>[0, 1]</code>&nbsp;的逆序对为&nbsp;<code>(0, 1)</code>&nbsp;。</li>\n</ul>\n</div>\n\n<div id=\"gtx-trans\" style=\"position: absolute; left: 198px; top: 756px;\">\n<div class=\"gtx-trans-icon\">&nbsp;</div>\n</div>\n\n<p>&nbsp;</p>\n\n<p><strong>提示：</strong></p>\n\n<ul>\n\t<li><code>2 &lt;= n &lt;= 300</code></li>\n\t<li><code>1 &lt;= requirements.length &lt;= n</code></li>\n\t<li><code>requirements[i] = [end<sub>i</sub>, cnt<sub>i</sub>]</code></li>\n\t<li><code>0 &lt;= end<sub>i</sub> &lt;= n - 1</code></li>\n\t<li><code>0 &lt;= cnt<sub>i</sub> &lt;= 400</code></li>\n\t<li>输入保证至少有一个&nbsp;<code>i</code>&nbsp;满足&nbsp;<code>end<sub>i</sub> == n - 1</code>&nbsp;。</li>\n\t<li>输入保证所有的&nbsp;<code>end<sub>i</sub></code>&nbsp;互不相同。</li>\n</ul>\n",
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                    "code": "class Solution {\npublic:\n    int numberOfPermutations(int n, vector<vector<int>>& requirements) {\n        \n    }\n};",
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                "Let <code>dp[i][j]</code> denote the number of arrays of length <code>i</code> with <code>j</code> inversions.",
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