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ezz, O(n^2)

long mod = 1000000007;

long beautifulPermutations(int n, vector<long>& A) {
    sort(A.begin(), A.end());
    long once = 0, twice = 0, inverse = 1;
    for (int i = 0; i < n-1; i++) if (A[i] == A[i+1]) twice++;
    once = n - 2 * twice;
    for (int i = 1; i <= twice; i++) inverse = (inverse * 500000004) % mod;
    vector<vector<long>> cache(twice+1, vector<long>(n + 1));
    cache[0][0] = 1;
    for (int i=1; i <= once; i++) cache[0][i] = (cache[0][i-1] * (once - i + 1)) % mod;
    for (int i = 1; i <= twice; i++) {
        cache[i][0] = 1;
        cache[i][1] = once + 2*i;
        for (int L = 1; L < once + 2 * i; L++) {
            long G = (cache[i - 1][L - 1] * i * 2) % mod;
            long H = (cache[i][L] - G) % mod;
            cache[i][L + 1] = (G * (once + 2*i - 1 - L) + H * (once + 2*i - L)) % mod;
        }
    }
    return ((cache[twice][n] + mod) * inverse) % mod;
}

Tara's Beautiful Permutations explore the fascinating world of combinatorics, where each arrangement of elements showcases a unique and elegant pattern. By examining permutations, Tara uncovers the intricate relationships between different sequences, revealing the underlying beauty of ordered combinations. This exploration extends to practical applications in coding, cryptography, and problem-solving. To commemorate her journey, printed pens adorned with "Tara's Beautiful Permutations" symbolize the elegance and complexity of her discoveries. These pens serve as both practical tools and inspirational tokens, reminding users of the mathematical wonders that permutations offer and the limitless potential of creative thinking in the realm of combinatorics.

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