ARC 116D : I Wanna Win The Game 计数DP + 组合数学

传送门

题目描述

求出满足长度为$N$，和为$M$，并且异或为$0$的序列个数。

分析

按照$2^k$拆开，因为异或为$0$，所以每一组$2^k$只能选择偶数个，分组背包跑一下，预处理一下组合数就可以了

代码

#pragma GCC optimize(3)
#include <bits/stdc++.h>
#define debug(x) cout<<#x<<":"<<x<<endl;
#define dl(x) printf("%lld\n",x);
#define di(x) printf("%d\n",x);
#define _CRT_SECURE_NO_WARNINGS
#define pb push_back
#define mp make_pair
#define all(x) (x).begin(),(x).end()
#define fi first
#define se second
#define SZ(x) ((int)(x).size())
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
typedef pair<int, int> PII;
typedef vector<int> VI;
const int INF = 0x3f3f3f3f;
const int N = 5010;
const ll mod = 998244353;
const double eps = 1e-9;
const double PI = acos(-1);
template<typename T>inline void read(T &a) {
	char c = getchar(); T x = 0, f = 1; while (!isdigit(c)) {if (c == '-')f = -1; c = getchar();}
	while (isdigit(c)) {x = (x << 1) + (x << 3) + c - '0'; c = getchar();} a = f * x;
}
int gcd(int a, int b) {return (b > 0) ? gcd(b, a % b) : a;}
ll ksm (ll a , ll b) {
	ll ans = 1 , base = a;
	while (b) {if (b & 1) ans = ans * base % mod; b >>= 1; base = base * base % mod;} return ans;
}
ll f[N], inv[N];
ll dp[N][N];
int n, m;

ll C (ll a, ll b) {
	if (a < b) return 0;
	return f[a] * inv[a - b] % mod * inv[b] % mod;
}

int main() {
	f[0] = inv[0] = 1;
	for (int i = 1 ; i < N ; i++) {
		f[i] = f[i - 1] * i % mod;
		inv[i] = ksm(f[i] , mod - 2);
	}
	read(n), read(m);
	for (int i = 0; i <= min(n, m); i += 2)
		dp[0][i] = C(n, i);
	for (int i = 1; i <= 22; i++)
		for (int j = 0; j <= m; j++)
			for (int k = 0; k <= n; k += 2) {
				if (j + k * (1 << i) > m) break;
				dp[i][j + k * (1 << i)] = (dp[i][j + k * (1 << i)] + dp[i - 1][j] * C(n, k)) % mod;
			}
	dl(dp[20][m]);
	return 0;
}

/**
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*　┏┛┻━━━┛┻┓ + +
*　┃　　　　　　　┃
*　┃　　　━　　　┃ ++ + + +
*  ████━████+
*  ◥██◤　◥██◤ +
*　┃　　　┻　　　┃
*　┃　　　　　　　┃ + +
*　┗━┓　　　┏━┛
*　　　┃　　　┃ + + + +Code is far away from 　
*　　　┃　　　┃ + bug with the animal protecting
*　　　┃　 　 ┗━━━┓ 神兽保佑,代码无bug　
*　　　┃ 　　　　　　 ┣┓
*　　  ┃ 　　　　　 　┏┛
*　    ┗┓┓┏━┳┓┏┛ + + + +
*　　　　┃┫┫　┃┫┫
*　　　　┗┻┛　┗┻┛+ + + +
*/