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<a href="/2019/06/16/20190616-My-Problems/" class="post-title-link" itemprop="url">我出的一些题目</a>
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<p>博主无聊时会出一些题目，难度不等。欢迎大家来玩！</p>
<p>注意：题目前带 <code>*</code> 的表示还没造完 / 已经投给某个比赛了，所以暂时不能查看。</p>
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<a href="/2019/12/12/20191212-Generating-Function-Notes/" class="post-title-link" itemprop="url">「学习笔记」生成函数入门</a>
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<h1 id="简介"><a href="#简介" class="headerlink" title="简介"></a>简介</h1><blockquote>
<p>在数学中，某个序列 $(a_n)_{n \in {\mathbb{N}}}$ 的母函数（又称生成函数，英语：Generating Function）是一种形式幂级数，其每一项的系数可以提供关于这个序列的信息。使用母函数解决问题的方法称为母函数方法。</p>
</blockquote>
<p>摘自维基百科。生成函数分为多种，本文将主要介绍普通生成函数（Ordinary Generating Function）和指数生成函数（Exponential Generating Function）。另外，本文还会提到大部分的多项式算法，用来解决生成函数的题目。</p>
<p>入门文章推荐：</p>
<ul>
<li><a href="https://zhuanlan.zhihu.com/p/52817010" target="_blank" rel="noopener">浅谈生成函数之 OGF</a></li>
<li><a href="https://zhuanlan.zhihu.com/p/53079223" target="_blank" rel="noopener">浅谈生成函数之 EGF</a></li>
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<a href="/2019/12/11/20191211-Polynomial-Algorithms-Notes/" class="post-title-link" itemprop="url">「学习笔记」多项式算法</a>
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<h1 id="前言"><a href="#前言" class="headerlink" title="前言"></a>前言</h1><p>在 OI 中，许多涉及到生成函数的计数题都需要使用一些多项式算法，所以掌握多项式算法是必要的。</p>
<h1 id="常见导数"><a href="#常见导数" class="headerlink" title="常见导数"></a>常见导数</h1><script type="math/tex; mode=display">
\begin {split}(x^n)' & = nx^{n - 1} \\(e^x)' & = e^x \\(a^x)' & = e^{x \ln a} \ln a \\(\ln x)' & = x^{-1}\end {split}</script>
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<a href="/2019/12/10/20191210-IOI2020-Homework-5/" class="post-title-link" itemprop="url">「集训队作业」IOI 2020 集训队作业小结 5</a>
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<h1 id="简介"><a href="#简介" class="headerlink" title="简介"></a>简介</h1><p>本文包括 IOI 2020 集训队作业 31 ~ 35 题的题解，下面是题单：</p>
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<li><a href="https://agc030.contest.atcoder.jp/tasks/agc030_d" target="_blank" rel="noopener">31.「AGC 030D」Inversion Sum</a></li>
<li><a href="https://agc030.contest.atcoder.jp/tasks/agc030_e" target="_blank" rel="noopener">32.「AGC 030E」Less than 3</a></li>
<li><a href="https://agc030.contest.atcoder.jp/tasks/agc030_f" target="_blank" rel="noopener">33.「AGC 030F」Permutation and Minimum</a></li>
<li><a href="https://agc031.contest.atcoder.jp/tasks/agc031_d" target="_blank" rel="noopener">34.「AGC 031D」A Sequence of Permutations</a></li>
<li><a href="https://agc031.contest.atcoder.jp/tasks/agc031_e" target="_blank" rel="noopener">35.「AGC 031E」Snuke the Phantom Thief</a></li>
<li><a href="https://agc031.contest.atcoder.jp/tasks/agc031_f" target="_blank" rel="noopener"><em>36.「AGC 031F」Walk on Graph</em></a></li>
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<a href="/2019/12/09/20191209-Probability-and-Expectation-1/" class="post-title-link" itemprop="url">「专题选做」概率与期望题目选做 1</a>
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<h1 id="TorusSailing"><a href="#TorusSailing" class="headerlink" title="TorusSailing"></a>TorusSailing</h1><p><a href="https://vjudge.net/problem/TopCoder-12984" target="_blank" rel="noopener">「Topcoder 12984」TorusSailing</a></p>
<h2 id="题目描述"><a href="#题目描述" class="headerlink" title="题目描述"></a>题目描述</h2><p>有 $n \times m$ 的网格，一人从 $(1, 1)$ 随机游走，每次等概率向下或向右，问走到 $(x, y)$ 的期望步数。</p>
<p>数据范围：$n, m \le 100$。</p>
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<p><a href="http://uoj.ac/problem/395" target="_blank" rel="noopener">「NOI 2018」你的名字（UOJ 395）</a></p>
<h1 id="题目描述"><a href="#题目描述" class="headerlink" title="题目描述"></a>题目描述</h1><p>给定字符串 $S$，$q$ 次询问给定 $(T, l, r)$，问 $T$ 有多少个本质不同的子串不在 $S$ 中出现。</p>
<p>数据范围：$\vert S \vert, \vert T \vert \le 5 \times 10^5, \sum \vert T \vert \le 10^6$。</p>
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<p><a href="https://codeforces.com/problemset/problem/547/E" target="_blank" rel="noopener">「Codeforces 547E」Mike and Friends</a></p>
<h1 id="题目描述"><a href="#题目描述" class="headerlink" title="题目描述"></a>题目描述</h1><p>给定 $n$ 个串，$q$ 次询问 $(l, r, k)$，问第 $k$ 个串在第 $[l, r]$ 个串中作为子串出现了多少次。</p>
<p>数据范围：$n \le 2 \times 10^5, q \le 5 \times 10^5$。</p>
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<h1 id="简介"><a href="#简介" class="headerlink" title="简介"></a>简介</h1><p>本文包括 IOI 2020 集训队作业 27 ~ 30 题的题解，下面是题单：</p>
<ul>
<li><a href="https://agc028.contest.atcoder.jp/tasks/agc028_e" target="_blank" rel="noopener"><em>25.「AGC 028E」High Elements</em></a></li>
<li><a href="https://agc028.contest.atcoder.jp/tasks/agc028_f" target="_blank" rel="noopener"><em>26.「AGC 028F」Reachable Cells</em></a></li>
<li><a href="https://agc029.contest.atcoder.jp/tasks/agc029_c" target="_blank" rel="noopener">27.「AGC 029C」Lexicographic constraints</a></li>
<li><a href="https://agc029.contest.atcoder.jp/tasks/agc029_e" target="_blank" rel="noopener">28.「AGC 029E」Wandering TKHS</a></li>
<li><a href="https://agc029.contest.atcoder.jp/tasks/agc029_f" target="_blank" rel="noopener">29.「AGC 029F」Construction of a tree</a></li>
<li><a href="https://agc030.contest.atcoder.jp/tasks/agc030_c" target="_blank" rel="noopener">30.「AGC 030C」Coloring Torus</a></li>
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<h1 id="Sharti"><a href="#Sharti" class="headerlink" title="Sharti"></a>Sharti</h1><p><a href="https://codeforces.com/contest/494/problem/E" target="_blank" rel="noopener">「Codeforces 494E」Sharti</a></p>
<h2 id="题目描述"><a href="#题目描述" class="headerlink" title="题目描述"></a>题目描述</h2><p>$n \times n$ 的网格，给定 $m$ 个矩形，它们的并是白色的，剩下的格子是黑色的。两人博弈，每次可以选择一个边长不超过 $k$ 的正方形满足它的右下角是白色，然后反转这个正方形的颜色。不能操作者输，问先手是否必胜。</p>
<p>数据范围：$m \le 5 \times 10^4, k \le n \le 10^9$。</p>
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<a href="/2019/12/03/20191203-Easy-Counting-1/" class="post-title-link" itemprop="url">「专题选做」简单计数题选做 1</a>
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<h1 id="SetAndSet"><a href="#SetAndSet" class="headerlink" title="SetAndSet"></a>SetAndSet</h1><p><a href="https://vjudge.net/problem/TopCoder-12004" target="_blank" rel="noopener">「Topcoder 12004」SetAndSet</a></p>
<h2 id="题目描述"><a href="#题目描述" class="headerlink" title="题目描述"></a>题目描述</h2><p>给 $n$ 个数 $a_1, a_2, \cdots, a_n$，要把它们划分成两个集合，使得两边的 And 一样，问方案数。</p>
<p>数据范围：$n \le 50, 0 \le a_i < 2^{20}$。</p>
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