luoguP1357 花园

首先对于\(n<10^5\)的范围,我们可以直接状压\(dp\).

设\(dp[i][s]\)表示\(dp\)到了第\(i\)位,最后\(m\)个状态的二进制压缩为\(s\)的方案数.

\(\therefore dp[i][s]=dp[i-1][\frac{s}{2}]+dp[i-1][(\frac{s}{2}+2^{m-1}]\).

但是这个题显然线性做法会超时.

于是使用套路:矩阵快速幂!

认真观察这个\(dp\)方程,我们可以发现对于任意一个合法状态的转移总是单一的,并且总是那个合法的\(bool\)矩阵.

于是我们可以对于中间状态\(T\)自乘\(n\)次,在对于每一个合法的初始状态,算出它的贡献,求和即可.

#pragma GCC optimize(3)
#include<bits/stdc++.h>
#define il inline
#define rg register
using namespace std;
typedef long long ll;
const int mod = 1e9 + 7, O = 55;
ll n;
int m, k, res, ans, sta[O];
struct Matrix {
	int a[O][O];
	Matrix() { memset(a, 0, sizeof a); }
	il int * operator [] (int x) { return a[x]; }
	il Matrix operator * (Matrix rhs) const {
		Matrix c;
		for (int k = 0; k <= res; ++k)
			for (int i = 0; i <= res; ++i)
				for (int j = 0; j <= res; ++j)
					(c[i][j] += 1ll * a[i][k] * rhs[k][j] % mod) %= mod;
		return c;
	}
}S, T, e;
il Matrix qpow(Matrix b, ll x) {
	Matrix Res = e;
	while (x) {
		if (x & 1) Res = Res * b;
		b = b * b; x >>= 1;
	}
	return Res;
}
int main() {
	scanf("%lld%d%d", &n, &m, &k);
	for (int i = 0; i < O; ++i) e[i][i] = 1;
	res = (1 << m) - 1;
	for (int s = 0; s <= res; ++s) {
		int tot = 0, num = s;
		while (num) tot += num & 1, num >>= 1;
		if (tot > k) continue;
		sta[s] = 1;
		T[s >> 1][s] = 1;
		if (tot != k || s & 1) T[(s >> 1) + (1 << m - 1)][s] = 1;
	}
	T = qpow(T, n);
	for (int s = 0; s <= res; ++s)
		if (sta[s]) {
			S[0][s] = 1;
			S = S * T;
			(ans += S[0][s]) %= mod;
			memset(S.a, 0, sizeof S.a);
		}
	printf("%d\n", ans);
	return 0;
}