Problem:
Score: 500 points

Problem Statement

Given is a permutation P of \{1, 2, \ldots, N\}.

For a pair (L, R) (1 \le L \lt R \le N), let X_{L, R} be the second largest value among P_L, P_{L+1}, \ldots, P_R.

Find \displaystyle \sum_{L=1}^{N-1} \sum_{R=L+1}^{N} X_{L,R}.

Constraints

2 \le N \le 10^5

1 \le P_i \le N

P_i \neq P_j (i \neq j)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
P_1 P_2 \ldots P_N

Output

Print \displaystyle \sum_{L=1}^{N-1} \sum_{R=L+1}^{N} X_{L,R}.

Sample Input 1

3
2 3 1

Sample Output 1

5

X_{1, 2} = 2, X_{1, 3} = 2, and X_{2, 3} = 1, so the sum is 2 + 2 + 1 = 5.

Sample Input 2

5
1 2 3 4 5

Sample Output 2

30

Sample Input 3

8
8 2 7 3 4 5 6 1

Sample Output 3

136

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

// #define DEBUGGING
#include <bits/stdc++.h>
using namespace std;
#define endl '\n'
#define ALL(V) (V).begin(), (V).end()
#define ALLR(V) (V).rbegin(), (V).rend()

template <typename T> using V = vector<T>;
template <typename T> using VV = V<V<T>>;
using ll = int64_t;
using ull = uint64_t;
using PLL = pair<ll, ll>;

template <typename T> const T& var_min(const T &t) { return t; }
template <typename T> const T& var_max(const T &t) { return t; }
template <typename T, typename... Tail> const T& var_min(const T &t, const Tail&... tail) { return min(t, var_min(tail...)); }
template <typename T, typename... Tail> const T& var_max(const T &t, const Tail&... tail) { return max(t, var_max(tail...)); }
template <typename T, typename... Tail> void chmin(T &t, const Tail&... tail) { t = var_min(t, tail...); }
template <typename T, typename... Tail> void chmax(T &t, const Tail&... tail) { t = var_max(t, tail...); }
template <typename T> const T& clamp(const T &t, const T &low, const T &high) { return max(low, min(high, t)); }
template <typename T> void chclamp(T &t, const T &low, const T &high) { t = clamp(t, low, high); }

namespace init__ {

struct InitIO {
    InitIO() {
        cin.tie(nullptr);
        ios_base::sync_with_stdio(false);
        cout << fixed << setprecision(30);
    }
} init_io;

}

#ifdef DEBUGGING
// #include "../debug/debug.cpp"
#include "../../debug/debug.cpp"
#else
#define DEBUG(...) 0
#define DEBUG_SEPARATOR_LINE 0
#endif

template <typename T>
T make_v(T init) { return init; }

template <typename T, typename... Tail>
auto make_v(T init, size_t s, Tail... tail) {
#define rec make_v(init, tail...)
    return V<decltype(rec)>(s, rec);
#undef rec
}

template <typename T>
class SegmentTree {
public:
    ll size;
    vector<T> node;
    function<T(T, T)> comp;
    T identity_ele;  //comp(x, identity_ele) == x

    SegmentTree(vector<T> v, function<T(T, T)> comp, T identity_ele) 
        : comp(comp),
          identity_ele(identity_ele)
    {
        ll tmp = 1;
        while(tmp < v.size()) tmp = (tmp << 1);
        this->size = tmp;
        this->identity_ele = identity_ele;
        this->comp = comp;
        node = V<T>(2 * size - 1, identity_ele);
        for(ll i = 0; i < v.size(); i++) {
            node[i + size - 1] = v[i];
        }
        for(ll i = size - 2; 0 <= i; i--) {
            node[i] = comp(node[i * 2 + 1], node[i * 2 + 2]);
        }
    }

    void update(ll pos, T value) {
        pos += size - 1;
        node[pos] = value;
        while(pos > 0) {
            pos = (pos - 1) / 2;
            node[pos] = comp(node[2 * pos + 1], node[2 * pos + 2]);
        }
    }

    T query(ll a, ll b) {
        return sub_query(a, b, 0, 0, size);
    }

    T sub_query(ll a, ll b, ll k, ll l, ll r) {
        if(r <= a || b <= l) {
            return identity_ele;
        } else if(a <= l && r <= b) {
            return node[k];
        } else {
            T left = sub_query(a, b, k * 2 + 1, l, (l + r) / 2);
            T right = sub_query(a, b, k * 2 + 2, (l + r) / 2, r);
            return comp(left, right);
        }
    }
};
    
ll N;

V<PLL> calc_range(V<ll> P, bool rev) {
    if(rev) reverse(ALL(P));
    const ll v_size = 1e5 + 10;
    SegmentTree<PLL> st(V<PLL>(v_size, PLL(-1, -1)),
                        [](PLL p, PLL q) {
                            ll a, b, c, d;
                            tie(a, b) = p;
                            tie(c, d) = q;
                            ll tmp[] = { a, b, c, d };
                            sort(tmp, tmp + 4);
                            return PLL(tmp[3], tmp[2]);
                        },
                        PLL(-1, -1));
    V<PLL> ret(P.size());
    for(ll i = 0; i < N; i++) {
        ll resf, ress;
        tie(resf, ress) = st.query(P[i], v_size);
        ll idx = i;
        if(rev) {
            resf = (N - 1) - resf;
            ress = (N - 1) - ress;
            idx = (N - 1) - i;
        }
        ret[idx].first = abs(idx - resf); 
        ret[idx].second = abs(ress - resf);
        st.update(P[i], PLL(i, -1));
    }
    return ret;
}

int main() {
    cin >> N;
    V<ll> P(N);
    for(auto &&ele : P) cin >> ele;
    auto left_cnt = calc_range(P, false);
    auto right_cnt = calc_range(P, true);
    ll ans = 0;
    for(ll i = 0; i < N; i++) {
        ll lf, ls, rf, rs;
        tie(lf, ls) = left_cnt[i];
        tie(rf, rs) = right_cnt[i];
        ll range = lf * rs + ls * rf;
        ans += range * P[i];
    }
    cout << ans << endl;
    return 0;
}