暑假第六测

今天是真的对我说再见

题解：

第一题：打表可以发现他走的路径是沿边缘的，m+1+ C(m+1, m) + C(m+2, m) + C(m+3, m) + …… + C(m+n, m) ( m > n)

而上面那一坨通过杨辉三角的性质可以发现： C(1, m) + C(2, m) + …… + C(m, m) = C(m+1, m+1) = 1;

C(1, m) + C(2, m) + ……  + C(m+n, m) = C(m+n+1, m+1);

#include <bits/stdc++.h>
#define ll long long
using namespace std;
const ll mod = 1e9 + 7;
const int M = 1e6 + 20;
ll ans;
void exgcd(ll a, ll b, ll &x, ll &y){
    if(!b){x = 1; y = 0; return ;}
    ll x0, y0;
    exgcd(b, a%b, x0, y0);
    x = y0;
    y = x0 - (a/b) * y0;
}

ll ni(ll a){
    ll x, y;
    exgcd(a, mod, x, y);
    return (x + mod) % mod;
}

void get2(ll m, ll n){
    ll now = 1, nni = 1;
    ll q = n + m + 1;
    for(register ll i = m+2; i <= q; i++) now = now * i % mod * ni(i - m - 1) % mod;
    ans = now; 
}


int main(){
    freopen("inverse.in","r",stdin);
    freopen("inverse.out","w",stdout);
    ll m, n;
    scanf("%I64d%I64d", &m, &n);
    if(m < n)swap(m, n);
    ans = 0;
    get2(m, n);
    printf("%I64d
", (ans + m) % mod);
}

以下是std

#include <cstdio>
#include <cctype>
#include <algorithm>
#define N_MAX 200000
#define M_MAX 50
#define lld "%I64u"
typedef unsigned long long lnt;
typedef unsigned unt;
typedef bool bnt;
const unt P = 1e9 + 7;
inline lnt moc(lnt a) { return a < P ? a : a - P; }
inline lnt mod(lnt a) { return a < P ? a : a % P; }
inline lnt inv(lnt a)
{
    lnt v = 1;
    while (a > 1) v = mod(v * (P - P / a)), a = P % a;
    return v;
}
const lnt v2 = inv(2);
struct buf
{
    operator int()
    {
        register int c = getchar(), x = 0;
        for (;!isdigit(c); c = getchar());
        for (; isdigit(c); c = getchar()) x = x * 10 - '0' + c;
        return x;
    }
} fio;
int n, m, i, j, k, a[N_MAX + 2], c[M_MAX + 1], p[M_MAX + 1], s[M_MAX + 1];
lnt w, b[N_MAX + 1], f[N_MAX + 1], ans;
inline bnt cmp(int x, int y) { return p[x] < p[y]; }
int main()
{
    freopen("crystal.in", "r", stdin);
    freopen("crystal.out", "w", stdout);
    n = fio, m = fio;
    for (i = 1; i <= m; ++i)
        c[i] = i;
    b[0] = 1;
    for (i = 1; i <= n; ++i)
        b[i] = mod(b[i - 1] * v2);
    for (i = 1; i <= n; ++i)
        f[i] = mod(f[i - 1] + b[i] * mod(mod(lnt(2 * i + 3) * i + 3) * i + 3));
    for (i = 1; i <= n; ++i)
        ++s[a[i] = fio];
    for (i = 1; i <= n; ++i)
    {
        p[a[i]] = i;
        for (j = 1; j <= m && c[j] != a[i]; ++j);
        while (j > 1 && p[c[j]] < p[c[j - 1]])
            std::swap(c[j], c[j - 1]), --j;
        for (k = 1; k <= m && !p[c[k]]; ++k);
        w = i < n ? v2 : 1;
        for (j = 1; j <= m; ++j)
            if (!p[j])
                w = mod(w * moc(1 - b[s[j] - (a[i + 1] == j)] + P));
        ans = mod(ans + w * moc(f[i] - f[i - 1] + P));
        w = mod(w * v2);
        for (j = k; j <= m; ++j)
        {
            ans = mod(ans + w * moc(f[i - p[c[j - 1]] - 1] - f[i - p[c[j]]] + P));
            if (j < m)
                ans = mod(ans + mod(w * moc(1 - b[s[c[j]] - 1 - (a[i + 1] == c[j])] + P)) * moc(f[i - p[c[j]]] - f[i - p[c[j]] - 1] + P));
            w = mod(w * moc(1 - b[s[c[j]] - (a[i + 1] == c[j])] + P));
        }
    }
    printf(lld "
", mod(ans * inv(b[n])));
    return 0;
}

以下是std

#include <cstdio>
#include <cmath>
#include <algorithm>
#define V_MAX 2
#define C_MAX 3
#define Q_MAX 20
#define N_MAX 100000
#define L_MAX 131072
#define R_MAX 17
typedef unsigned long long lnt;
typedef unsigned unt;
typedef void vnt;
typedef double dec;
const unt P = 47;
const dec tau = std::acos(dec(-1)) * 2;
inline unt mod(unt a) { return a % P; }
inline unt moc(unt a) { return a < P ? a : a - P; }
inline vnt upd(unt & a, unt b) { a = mod(a + b); }
inline vnt upc(unt & a, unt b) { a = moc(a + b); }
struct vec
{
    dec x, y;
    vec operator + (const vec & z) const { return (vec) {x + z.x, y + z.y}; }
    vec operator - (const vec & z) const { return (vec) {x - z.x, y - z.y}; }
    vec operator * (const vec & z) const { return (vec) {x * z.x - y * z.y, x * z.y + y * z.x}; }
    friend vec operator * (unt w, const vec & z) { return (vec) {w * z.x, w * z.y}; }
} vec_0, vec_1 = (vec) {1, 0}, w[R_MAX];
int L, R, rev[L_MAX + 1];
inline vnt fft(vec a[])
{
    int i, j, k, r = R; vec x, u, v;
    for (i = 1; i < L; ++i)
        if (i < rev[i])
            std::swap(a[i], a[rev[i]]);
    for (k = 1, --r; k < L; k <<= 1, --r)
        for (i = 0; i < L; i += k << 1)
            for (j = 0, x = vec_1; j < k; ++j, x = x * w[r])
                u = a[i + j], v = x * a[i + j + k], a[i + j] = u + v, a[i + j + k] = u - v;
}
int V, d[V_MAX][C_MAX];
unt e[V_MAX][C_MAX], f[V_MAX][V_MAX][N_MAX + 1];
vec a[V_MAX][V_MAX][L_MAX + 1], b[V_MAX][V_MAX][L_MAX + 1], c[V_MAX][V_MAX][L_MAX + 1];
inline vnt sol(int l, int r)
{
    if (r - l == 1)
    {
        if (l == 0)
            for (int x = 0; x < V; ++x)
                f[x][x][0] = 1;
        for (int x = 0; x < V; ++x)
            for (int y = 0; y < V; ++y)
                upd(f[x][y][l + 1], e[x][2] * f[d[x][2]][y][l]);
    }
    else
    {
        int m = l + ((r - l) >> 1);
        sol(l, m);
        if (l == 0)
        {
            for (L = 1, R = 0; L < (m - l) * 2 - 1; L <<= 1, ++R);
            for (int i = 1; i < L; ++i)
                rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (R - 1));
            w[0] = (vec) {std::cos(tau / L), std::sin(tau / L)};
            for (int i = 0; i < R - 1; ++i)
                w[i + 1] = w[i] * w[i];
            for (int x = 0; x < V; ++x)
                for (int y = 0; y < V; ++y)
                {
                    for (int i = 0; i < m; ++i)
                        a[x][y][i].x = f[x][y][i], a[x][y][i].y = 0;
                    for (int i = m; i < L; ++i)
                        a[x][y][i] = vec_0;
                    fft(a[x][y]);
                }
            for (int x = 0; x < V; ++x)
                for (int y = 0; y < V; ++y)
                    for (int z = 0; z < V; ++z)
                        for (int i = 0; i < L; ++i)
                            c[x][z][i] = c[x][z][i] + mod(e[x][0] * e[y][1]) * a[d[x][0]][y][i] * a[d[y][1]][z][i];
        }
        else
        {
            for (L = 1, R = 0; L < m - l + r - l - 2; L <<= 1, ++R);
            for (int i = 1; i < L; ++i)
                rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (R - 1));
            w[0] = (vec) {std::cos(tau / L), std::sin(tau / L)};
            for (int i = 0; i < R - 1; ++i)
                w[i + 1] = w[i] * w[i];
            for (int x = 0; x < V; ++x)
                for (int y = 0; y < V; ++y)
                {
                    for (int i = l; i < m; ++i)
                        a[x][y][i - l].x = f[x][y][i], a[x][y][i - l].y = 0;
                    for (int i = m - l; i < L; ++i)
                        a[x][y][i] = vec_0;
                    fft(a[x][y]);
                }
            for (int x = 0; x < V; ++x)
                for (int y = 0; y < V; ++y)
                {
                    for (int i = 0; i < r - l - 1; ++i)
                        b[x][y][i].x = f[x][y][i], b[x][y][i].y = 0;
                    for (int i = r - l - 1; i < L; ++i)
                        b[x][y][i] = vec_0;
                    fft(b[x][y]);
                }
            for (int x = 0; x < V; ++x)
                for (int y = 0; y < V; ++y)
                    for (int z = 0; z < V; ++z)
                        for (int i = 0; i < L; ++i)
                            c[x][z][i] = c[x][z][i] + mod(e[x][0] * e[y][1]) * (a[d[x][0]][y][i] * b[d[y][1]][z][i] + b[d[x][0]][y][i] * a[d[y][1]][z][i]);
        }
        w[0].y = -w[0].y;
        for (int i = 0; i < R - 1; ++i)
            w[i + 1] = w[i] * w[i];
        for (int u = 0; u < V; ++u)
            for (int v = 0; v < V; ++v)
            {
                fft(c[u][v]);
                for (int i = m + 1; i < r + 1; ++i)
                    upd(f[u][v][i], unt(lnt(c[u][v][i - l - 2].x / L + 0.5) % P));
                for (int i = 0; i < L; ++i)
                    c[u][v][i] = vec_0;
            }
        sol(m, r);
    }
}
int Q, N;
struct qry { int s, t, n; } q[Q_MAX];
int main()
{
    freopen("bracket.in", "r", stdin);
    freopen("bracket.out", "w", stdout);
    scanf("%d", &V);
    for (int x = 0; x < V; ++x)
        for (int k = 0; k < C_MAX; ++k)
            scanf("%d %u", &d[x][k], &e[x][k]), e[x][k] %= P;
    scanf("%d", &Q);
    for (int k = 0; k < Q; ++k)
        scanf("%d %d %d", &q[k].s, &q[k].t, &q[k].n), N = std::max(N, q[k].n);
    sol(0, N);
    for (int k = 0; k < Q; ++k)
        printf("%u
", f[q[k].s][q[k].t][q[k].n]);
    return 0;
}