比较套路的组合数学题 For a sequence a of n integers between 1 and m, inclusive, denote f(a) as the number of distinct subsequences of a (including the empty subsequence). You are given two positive integers n and m. Let S be the set of all sequences of length …

For a sequence a of n integers between 1 and m, inclusive, denote f(a) as the number of distinct subsequences of a (including the empty subsequence). You are given two positive integers n and m. Let S be the set of all sequences of length n consistin…

[CF660E]Different Subsets For All Tuples 题意:对于所有长度为n,每个数为1,2...m的序列,求出每个序列的本质不同的子序列的数目之和.(多个原序列可以有相同的子序列) $n,m\le 10^6$ 题解:结论:一个子序列出现的次数只与其长度有关. 我们可以分别求出每种长度的子序列出现的总次数,显然答案为: $\sum\limits_{i=1}^nm^i\sum\limits_{j=i}^nC_{j-1}^{i-1}(m-1)^{j-i}m^{n-j}$…

E. Different Subsets For All Tuples 题目连接: http://www.codeforces.com/contest/660/problem/E Description For a sequence a of n integers between 1 and m, inclusive, denote f(a) as the number of distinct subsequences of a (including the empty subsequence)…

点此看题面 大致题意: 有一个长度为\(n\)的数列,每个位置上数字的值在\([1,m]\)范围内,则共有\(m^n\)种可能的数列.分别求出每个数列中本质不同的子序列个数,然后求和. 一些分析 首先,我们单独考虑空序列的个数\(m^n\),然后接下来就可以只考虑非空序列的个数了. 假设有一个长度为\(i\)的子序列(\(1\le i\le n\)),且其在序列中的位置分别为\(pos_1,pos_2,...,pos_i\),值分别为\(val_1,val_2,...,val_i\). 则我们强…

看了官方题解+q神的讲解才懂... 智商问题.. 讲道理..数学真的比脱单难啊... 题目链接: http://codeforces.com/problemset/problem/660/E 题意: 给定数字范围,问由这些数字组成的长度为n的串的子序列有多少种? 分析: 方法一: 枚举长度k,计算以其为子序列的原串种数. k=0时,ans=mn k≥1时,设序列元素为x1,x2,x3...xk,为了避免重复,我们假设当前位置是第一次出现xi,即要求x1之前的元素不会出现x1,x1和x2之间的元素…

大意: 定义$f(a)$表示序列$a$本质不同子序列个数. 给定$n,m$, 求所有长$n$元素范围$[1,m]$的序列的$f$值之和. 显然长度相同的子序列贡献是相同的. 不考虑空串, 假设长$x$, 枚举第一次出现位置, 可以得到贡献为$\sum\limits_{i=x}^n\binom{i-1}{x-1}(m-1)^{i-x}m^{n-i}$ 总的答案就为$\sum\limits_{x=1}^n m^x \sum\limits_{i=x}^n\binom{i-1}{x-1}(m-1)^{i…

因为垃圾电脑太卡了就重开了一个... 前传:多项式Ⅰ u1s1 我预感还会有Ⅲ 多项式基础操作: 例题: 26. CF438E The Child and Binary Tree 感觉这题作为第一题还蛮合适的( 首先我们设 \(f_i\) 为权值之和为 \(i\) 的符合要求的二叉树的个数. 显然可以枚举根节点的权值.左子树的权值之和进行转移. 也就是 \(f_i=\sum\limits_{x\in S}\sum\limits_{y=0}^{i-S}f_yf_{i-x-y}\) 如果我们记 \(…

得分: \(100+100+0=200\)(\(T1\)在最后\(2\)分钟写了出来,\(T2\)在最后\(10\)分钟写了出来,反而\(T3\)写了\(4\)个小时爆\(0\)) \(T1\):风王结界 一道数学题,看完题解是无比的简单. 比赛时最后两分钟把式子凑出来了(\(RP\)爆表)... 题解详见这篇博客:[CF660E]Different Subsets For All Tuples(组合数学),即这题模数为\(10^9+7\)的版本. 下面直接给出代码: #include<bits…

C. Square Subsets time limit per test 4 seconds memory limit per test 256 megabytes input standard input output standard output Petya was late for the lesson too. The teacher gave him an additional task. For some array a Petya should find the number…

Given a collection of integers that might contain duplicates, nums, return all possible subsets. Note: The solution set must not contain duplicate subsets. For example,If nums = [1,2,2], a solution is: [ [2], [1], [1,2,2], [2,2], [1,2], [] ] 分析: 因为有重…

Given a set of distinct integers, nums, return all possible subsets. Note: The solution set must not contain duplicate subsets. For example,If nums = [1,2,3], a solution is: [ [3], [1], [2], [1,2,3], [1,3], [2,3], [1,2], [] ] 分析:  用位操作非常简单,对于n个数来说共有2…

Given a collection of integers that might contain duplicates, S, return all possible subsets. Note: Elements in a subset must be in non-descending order. The solution set must not contain duplicate subsets. For example,If S = [1,2,2], a solution is:…

Given a set of distinct integers, S, return all possible subsets. Note: Elements in a subset must be in non-descending order. The solution set must not contain duplicate subsets. For example,If S = [1,2,3], a solution is: [ [3], [1], [2], [1,2,3], [1…

iOS Swift-元组tuples(The Swift Programming Language) 什么是元组? 元组(tuples)是把多个值组合成一个复合值,元组内的值可以使任意类型,并不要求是相同类型. 元组长什么样? 如下: let nameAndAge = ("旭宝爱吃鱼",22); print(nameAndAge); 打印出了什么呢? 如下: ("旭宝爱吃鱼", 22) 是不是对元组有了清晰的了解了呢... 那么下面继续深入了解一下. 元组的分解 比…

求集合的所有子集问题 LeetCode:Subsets Given a set of distinct integers, S, return all possible subsets. Note: Elements in a subset must be in non-descending order. The solution set must not contain duplicate subsets. For example,If S = [1,2,3], a solution is:…

5.3 Tuples and Sequences We saw that lists and strings have many common properties, e.g., indexing and slicing operations. They are two examples of sequence data types. Since Python is an evolving language, other sequence data types may be added. The…

Given a collection of integers that might contain duplicates, S, return all possible subsets. Note: Elements in a subset must be in non-descending order. The solution set must not contain duplicate subsets. For example,If S = [1,2,2], a solution is:…

Given a set of distinct integers, S, return all possible subsets. Note: Elements in a subset must be in non-descending order. The solution set must not contain duplicate subsets. 就是找出数字的全部子集. 我的思路: 设输入是 1  2  3  4 先把输入从小到大排序 长度 0 : [] 长度 1 : 把输入数据从第一…

题目链接 题意:有一排砖,可以染红蓝绿黄四种不同的颜色,要求红和绿两种颜色砖的个数都是偶数,问一共有多少种方案,结果对10007取余. 题解:刚看这道题第一感觉是组合数学,正向推了一会还没等推出来队友就打表找到公式了,然后我就写了一个快速幂加个费马小定理就过了去看别的题了,赛后找到了一个很不错的博客:传送门,原来这道题也可以用DP+矩阵快速幂AC.下面说下组合数学的做法: 首先一共有4^n种情况,我们减去不符合条件的情况就行了,从中取k个进行染红绿色一共C(n,k)种情况,剩下的蓝黄色一共有2^…

Given a collection of integers that might contain duplicates, S, return all possible subsets. Note: Elements in a subset must be in non-descending order. The solution set must not contain duplicate subsets. For example,If S = [1,2,2], a solution is:…

Subsets Given a set of distinct integers, S, return all possible subsets. Note: Elements in a subset must be in non-descending order. The solution set must not contain duplicate subsets. For example, If S = [1,2,3], a solution is: [ [3], [1], [2], [1…

[construction of tuples containing 0 or 1 items] the syntax has some extra quirks to accommodate these. Empty tuples are constructed by an empty pair of parentheses; a tuple with one item is constructed by following a value with a comma (it is not su…

子集系列问题: Coding 问题中有时会出现这样的问题:给定一个集合,求出这个集合所有的子集(所谓子集,就是包含原集合中的一部分元素的集合). 或者求出满足一定要求的子集,比如子集中元素总和为定值,子集元素个数为定值等等. 我把它们归类为子集系列问题. leetcode上原题链接: 思路分析: 思路一 可以用递推的思想,观察S=[], S =[1], S = [1, 2] 时解的变化. 可以发现S=[1, 2] 的解就是 把S = [1]的所有解末尾添上2,然后再并上S = [1]里面的原有解…

below is a good answer for this question , so I copy on here for some people need it By the way, the three forms can be combined as follows: def f(a,*b,**c): All single arguments beyond the first will end up with the tuple b, and all key/value argume…

9.4 Write a method to return all subsets of a set. LeetCode上的原题,请参见我之前的博客Subsets 子集合和Subsets II 子集合之二. 解法一: class Solution { public: vector<vector<int> > getSubsets(vector<int> &S) { vector<vector<); ; i < S.size(); ++i) { i…

模板问题: 1. 取物品 (comb.pas/c/cpp) [问题描述] 现在有n个物品(有可能相同),请您编程计算从中取k个有多少种不同的取法.[输入] 输入文件有两行,第一行包含两个整数n,k(2<=n<=30,0<=k<=n).第二行,包含n个整数表示物品的编号(范围1..1000).编号相同的物品看作同一种物品. 想想看,组合数学中有这样的模型吗?答案是,肯定会有的.但是因为这个问题的灵活性,还没有通项公式. 想象当每个数都不同时,这个问题就变成了${n \choose k…

Subsets II Given a collection of integers that might contain duplicates, S, return all possible subsets. Note: Elements in a subset must be in non-descending order. The solution set must not contain duplicate subsets. For example,If S = [1,2,2], a so…

Subsets Given a set of distinct integers, S, return all possible subsets. Note: Elements in a subset must be in non-descending order. The solution set must not contain duplicate subsets. For example,If S = [1,2,3], a solution is: [ [3], [1], [2], [1,…

Given a set of distinct integers, nums, return all possible subsets. Note: Elements in a subset must be in non-descending order. The solution set must not contain duplicate subsets. For example, If nums = [1,2,3], a solution is: [ [3], [1], [2], [1,2…