結果

問題 No.1193 Penguin Sequence
コンテスト
ユーザー lam6er
提出日時 2025-04-09 20:56:08
言語 PyPy3
(7.3.15)
結果
AC  
実行時間 699 ms / 2,000 ms
コード長 3,543 bytes
コンパイル時間 372 ms
コンパイル使用メモリ 82,208 KB
実行使用メモリ 167,368 KB
最終ジャッジ日時 2025-04-09 20:56:56
合計ジャッジ時間 19,860 ms
ジャッジサーバーID
（参考情報） 		judge3 / judge5
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ファイルパターン 結果
sample AC * 3
other AC * 38
権限があれば一括ダウンロードができます

ソースコード

diff # 

import bisect

MOD = 998244353

def main():
    import sys
    n, *rest = list(map(int, sys.stdin.read().split()))
    A = rest[:n]

    # Compute cnt_all_pairs and cnt_inversion_pairs using Fenwick Tree

    # Coordinate compression for all elements
    sorted_unique = sorted(list(set(A)))
    rank = {v:i for i, v in enumerate(sorted_unique)}
    m = len(sorted_unique)
    freq = [0] * m
    for v in A:
        idx = bisect.bisect_left(sorted_unique, v)
        freq[idx] += 1

    # Compute cnt_all_pairs
    class FenwickTree:
        def __init__(self, size):
            self.n = size
            self.tree = [0] * (self.n + 1)
        
        def update(self, idx, delta):
            while idx <= self.n:
                self.tree[idx] += delta
                idx += idx & -idx
        
        def query(self, idx):
            res = 0
            while idx > 0:
                res += self.tree[idx]
                idx -= idx & -idx
            return res
        
        def range_query(self, l, r):
            return self.query(r) - self.query(l - 1)

    ft_all = FenwickTree(m)
    for i in range(m):
        ft_all.update(i + 1, freq[i])  # 1-based indexing
    
    cnt_all_pairs = 0
    for v in A:
        idx = bisect.bisect_right(sorted_unique, v)
        if idx < m:
            count = ft_all.range_query(idx + 1, m)  # 1-based to m
            cnt_all_pairs += count
    cnt_all_pairs %= MOD

    # Compute cnt_inversion_pairs (normal inversion count)
    # Coordinate compression for the original array for inversion count
    sorted_A = sorted(list(set(A)))
    rank_inv = {v:i+1 for i, v in enumerate(sorted_A)}  # 1-based
    max_rank = len(sorted_A)
    ft_inv = FenwickTree(max_rank)
    inv_count = 0
    for j in reversed(range(n)):
        v_rank = bisect.bisect_left(sorted_A, A[j])
        v_rank += 1  # 1-based
        inv_count += ft_inv.query(v_rank - 1)
        ft_inv.update(v_rank, 1)
    inv_count %= MOD

    # Precompute factorials and inverse factorials
    max_fact = n
    fact, inv_fact = [1]*(max_fact + 1), [1]*(max_fact + 1)
    for i in range(1, max_fact + 1):
        fact[i] = fact[i-1] * i % MOD
    inv_fact[max_fact] = pow(fact[max_fact], MOD-2, MOD)
    for i in range(max_fact-1, -1, -1):
        inv_fact[i] = inv_fact[i+1] * (i+1) % MOD

    def comb(n_, k_):
        if k_ < 0 or k_ > n_:
            return 0
        return fact[n_] * inv_fact[k_] % MOD * inv_fact[n_ - k_] % MOD

    # Compute T = product of C(n,i) for i in 1..n
    T = 1
    for i in range(1, n+1):
        c = comb(n, i)
        T = T * c % MOD

    # Compute S = sum_{k<l} k*l = (s1^2 - s2) / 2
    s1 = n * (n + 1) // 2
    s1_mod = s1 % MOD
    s2 = n * (n + 1) * (2 * n + 1) // 6
    s2_mod = s2 % MOD
    S_mod = (pow(s1_mod, 2, MOD) - s2_mod) * pow(2, MOD-2, MOD) % MOD

    # Compute sum_k_part = sum_{k=2 to n} k(k-1) = (s2 - s1) % MOD
    sum_k_part = (s2 - s1) % MOD

    # Compute denominators and their inverses
    n_mod = n % MOD
    inv_n = pow(n_mod, MOD-2, MOD)
    inv_n_square = pow(n_mod * n_mod % MOD, MOD-2, MOD)
    if n >= 2:
        inv_n_times_n_minus_1 = pow(n_mod * ( (n-1) % MOD ) % MOD, MOD-2, MOD)
    else:
        inv_n_times_n_minus_1 = 0  # but for n<2, sum_k_part is 0, so term2 is 0

    term1 = cnt_all_pairs * S_mod % MOD
    term1 = term1 * inv_n_square % MOD

    term2 = inv_count * sum_k_part % MOD
    term2 = term2 * inv_n_times_n_minus_1 % MOD

    ans = (term1 + term2) % MOD
    ans = ans * T % MOD

    print(ans)

if __name__ == '__main__':
    main()
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