P4608

计数dp #数学 #组合 #去重

最短路径 \(\rightarrow\) 只向右下走

考虑 \(dp_{i,j}\) 表示到 \(i\) 号点，填了 \(j\) 个方案数

注意不可以直接从所有子节点加过来，会重复计算，要把当前点经过另一个点再到目标点的方案减去

// Author: xiaruize
#ifndef ONLINE_JUDGE
bool start_of_memory_use;
#else
#define debug(x)
#endif
#include <bits/stdc++.h>
using namespace std;
#ifndef ONLINE_JUDGE
clock_t start_clock = clock();
#endif
namespace __DEBUG_UTIL__
{
    using namespace std;
    /* Primitive Datatypes Print */
    void print(const char *x) { cerr << x; }
    void print(bool x) { cerr << (x ? "T" : "F"); }
    void print(char x) { cerr << '\'' << x << '\''; }
    void print(signed short int x) { cerr << x; }
    void print(unsigned short int x) { cerr << x; }
    void print(signed int x) { cerr << x; }
    void print(unsigned int x) { cerr << x; }
    void print(signed long int x) { cerr << x; }
    void print(unsigned long int x) { cerr << x; }
    void print(signed long long int x) { cerr << x; }
    void print(unsigned long long int x) { cerr << x; }
    void print(float x) { cerr << x; }
    void print(double x) { cerr << x; }
    void print(long double x) { cerr << x; }
    void print(string x) { cerr << '\"' << x << '\"'; }
    template <size_t N>
    void print(bitset<N> x) { cerr << x; }
    void print(vector<bool> v)
    { /* Overloaded this because stl optimizes vector<bool> by using
          _Bit_reference instead of bool to conserve space. */
        int f = 0;
        cerr << '{';
        for (auto &&i : v)
            cerr << (f++ ? "," : "") << (i ? "T" : "F");
        cerr << "}";
    }
    /* Templates Declarations to support nested datatypes */
    template <typename T>
    void print(T &&x);
    template <typename T>
    void print(vector<vector<T>> mat);
    template <typename T, size_t N, size_t M>
    void print(T (&mat)[N][M]);
    template <typename F, typename S>
    void print(pair<F, S> x);
    template <typename T, size_t N>
    struct Tuple;
    template <typename T>
    struct Tuple<T, 1>;
    template <typename... Args>
    void print(tuple<Args...> t);
    template <typename... T>
    void print(priority_queue<T...> pq);
    template <typename T>
    void print(stack<T> st);
    template <typename T>
    void print(queue<T> q);
    /* Template Datatypes Definitions */
    template <typename T>
    void print(T &&x)
    {
        /*  This works for every container that supports range-based loop
            i.e. vector, set, map, oset, omap, dequeue */
        int f = 0;
        cerr << '{';
        for (auto &&i : x)
            cerr << (f++ ? "," : ""), print(i);
        cerr << "}";
    }
    template <typename T>
    void print(vector<vector<T>> mat)
    {
        int f = 0;
        cerr << "\n~~~~~\n";
        for (auto &&i : mat)
        {
            cerr << setw(2) << left << f++, print(i), cerr << "\n";
        }
        cerr << "~~~~~\n";
    }
    template <typename T, size_t N, size_t M>
    void print(T (&mat)[N][M])
    {
        int f = 0;
        cerr << "\n~~~~~\n";
        for (auto &&i : mat)
        {
            cerr << setw(2) << left << f++, print(i), cerr << "\n";
        }
        cerr << "~~~~~\n";
    }
    template <typename F, typename S>
    void print(pair<F, S> x)
    {
        cerr << '(';
        print(x.first);
        cerr << ',';
        print(x.second);
        cerr << ')';
    }
    template <typename T, size_t N>
    struct Tuple
    {
        static void printTuple(T t)
        {
            Tuple<T, N - 1>::printTuple(t);
            cerr << ",", print(get<N - 1>(t));
        }
    };
    template <typename T>
    struct Tuple<T, 1>
    {
        static void printTuple(T t) { print(get<0>(t)); }
    };
    template <typename... Args>
    void print(tuple<Args...> t)
    {
        cerr << "(";
        Tuple<decltype(t), sizeof...(Args)>::printTuple(t);
        cerr << ")";
    }
    template <typename... T>
    void print(priority_queue<T...> pq)
    {
        int f = 0;
        cerr << '{';
        while (!pq.empty())
            cerr << (f++ ? "," : ""), print(pq.top()), pq.pop();
        cerr << "}";
    }
    template <typename T>
    void print(stack<T> st)
    {
        int f = 0;
        cerr << '{';
        while (!st.empty())
            cerr << (f++ ? "," : ""), print(st.top()), st.pop();
        cerr << "}";
    }
    template <typename T>
    void print(queue<T> q)
    {
        int f = 0;
        cerr << '{';
        while (!q.empty())
            cerr << (f++ ? "," : ""), print(q.front()), q.pop();
        cerr << "}";
    }
    /* Printer functions */
    void printer(const char *) {} /* Base Recursive */
    template <typename T, typename... V>
    void printer(const char *names, T &&head, V &&...tail)
    {
        /* Using && to capture both lvalues and rvalues */
        int i = 0;
        for (size_t bracket = 0; names[i] != '\0' and (names[i] != ',' or bracket != 0); i++)
            if (names[i] == '(' or names[i] == '<' or names[i] == '{')
                bracket++;
            else if (names[i] == ')' or names[i] == '>' or names[i] == '}')
                bracket--;
        cerr.write(names, i) << " = ";
        print(head);
        if (sizeof...(tail))
            cerr << " ||", printer(names + i + 1, tail...);
        else
            cerr << "]\n";
    }
    /* PrinterArr */
    void printerArr(const char *) {} /* Base Recursive */
    template <typename T, typename... V>
    void printerArr(const char *names, T arr[], size_t N, V... tail)
    {
        size_t ind = 0;
        for (; names[ind] and names[ind] != ','; ind++)
            cerr << names[ind];
        for (ind++; names[ind] and names[ind] != ','; ind++)
            ;
        cerr << " = {";
        for (size_t i = 0; i < N; i++)
            cerr << (i ? "," : ""), print(arr[i]);
        cerr << "}";
        if (sizeof...(tail))
            cerr << " ||", printerArr(names + ind + 1, tail...);
        else
            cerr << "]\n";
    }
}
#ifndef ONLINE_JUDGE
#define debug(...) std::cerr << __LINE__ << ": [", __DEBUG_UTIL__::printer(#__VA_ARGS__, __VA_ARGS__)
#define debugArr(...) std::cerr << __LINE__ << ": [", __DEBUG_UTIL__::printerArr(#__VA_ARGS__, __VA_ARGS__)
#else
#define debug(...)
#define debugArr(...)
#endif
#define int long long
#define ull unsigned long long
#define ALL(a) (a).begin(), (a).end()
#define pb push_back
#define mk make_pair
#define pii pair<int, int>
#define pis pair<int, string>
#define sec second
#define fir first
#define sz(a) int((a).size())
#define Yes cout << "Yes" << endl
#define YES cout << "YES" << endl
#define No cout << "No" << endl
#define NO cout << "NO" << endl
#define mms(arr, n) memset(arr, n, sizeof(arr))
#define rep(i, a, n) for (int i = (a); i <= (n); ++i)
#define per(i, n, a) for (int i = (n); i >= (a); --i)
int max(int a, int b)
{
    if (a > b)
        return a;
    return b;
}
int min(int a, int b)
{
    if (a < b)
        return a;
    return b;
}
const int INF = 0x3f3f3f3f3f3f3f3f;
const int MOD = 1e9 + 9;
const int N = 2e6 + 10;

int fac[2000009], inv[2000009];
int qpow(int a, int b)
{
    int ans = 1;
    while (b)
    {
        if (b & 1)
            ans = (ans * a) % MOD;
        a = (a * a) % MOD;
        b /= 2;
    }
    return ans;
}

void init()
{
    fac[0] = 1;
    for (int i = 1; i <= 2000000; i++)
        fac[i] = (fac[i - 1] * i) % MOD;
    inv[2000000] = qpow(fac[2000000], MOD - 2);
    for (int i = 2000000 - 1; i >= 0; i--)
        inv[i] = (inv[i + 1] * (i + 1)) % MOD;
}

int C(int x, int y)
{
    if (x < 0 || y < 0 || x < y)
        return 0;
    if (x == y || y == 0)
        return 1;
    return fac[x] * inv[x - y] % MOD * inv[y] % MOD;
}

int calc(int x, int y, int xx, int yy, int k)
{
    if (x > xx || y > yy)
        return 0;
    if (x == xx && y == yy)
        return k == 0;
    if (x == xx)
        return C(yy - y - 1, k - 1);
    if (y == yy)
        return C(xx - x - 1, k - 1);
    int res = 0;
    rep(i, 1, k - 1)
    {
        (res += C(k, i) * C(xx - x - 1, i - 1) % MOD * C(yy - y - 1, k - i - 1) % MOD) %= MOD;
        res = (res % MOD + MOD) % MOD;
    }
    return res;
}

int n, dx, dy, k;
struct node
{
    int x, y, val;
} s[N];
int dp[45][15];

bool cmp(node x, node y)
{
    if (x.x != y.x)
        return x.x < y.x;
    return x.y < y.y;
}

void solve()
{
    cin >> n;
    rep(i, 2, n + 1) cin >> s[i].x;
    cin >> n;
    rep(i, 2, n + 1) cin >> s[i].y;
    cin >> n;
    rep(i, 2, n + 1) cin >> s[i].val;
    cin >> dx >> dy >> k;
    s[1] = {0, 0, 0};
    n++;
    s[++n] = {dx, dy, 0};
    sort(s + 1, s + n + 1, cmp);
    while (s[n].x != dx || s[n].y != dy)
        n--;
    debug(s[n].x, s[n].y, s[n].val, n);
    dp[1][0] = 1;
    rep(i, 1, n)
    {
        rep(j, 1, i - 1)
        {
            if (s[i].x >= s[j].x && s[i].y >= s[j].y)
            {
                rep(a, 0, k)
                {
                    rep(b, 0, a)
                    {
                        dp[i][a] = ((dp[i][a] + dp[j][a - b] * calc(s[j].x + s[j].val, s[j].y, s[i].x, s[i].y, b) % MOD) % MOD + MOD) % MOD;
                        if (s[j].val)
                        {
                            dp[i][a] = ((dp[i][a] + dp[j][a - b] * calc(s[j].x, s[j].y + s[j].val, s[i].x, s[i].y, b) % MOD) % MOD + MOD) % MOD;
                            dp[i][a] = ((dp[i][a] - dp[j][a - b] * calc(s[j].x, s[j].y, s[i].x, s[i].y, b) % MOD) % MOD + MOD) % MOD;
                        }
                        dp[i][a] = (dp[i][a] % MOD + MOD) % MOD;
                    }
                    // debug(dp[i][a]);
                }
            }
        }
    }
    int res = 0;
    rep(i, 1, k) res = (res + dp[n][i]) % MOD;
    cout << res << endl;
}

#ifndef ONLINE_JUDGE
bool end_of_memory_use;
#endif

signed main()
{
    // freopen(".in","r",stdin);
    // freopen(".out","w",stdout);
    ios::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);
    int testcase = 1;
    init();
    // cin >> testcase;
    while (testcase--)
        solve();
#ifndef ONLINE_JUDGE
    cerr << "Memory use:" << (&end_of_memory_use - &start_of_memory_use) / 1024.0 / 1024.0 << "MiB" << endl;
    cerr << "Time use:" << (double)clock() / CLOCKS_PER_SEC * 1000.0 << "ms" << endl;
#endif
    return 0;
}