Find the Index of Permutation — C++

// Time:  O(nlogn)
// Space: O(n)

// bit, fenwick tree, combinatorics
class Solution {
private:
    static const uint32_t MOD = 1e9 + 7;

public:
    int getPermutationIndex(vector<int>& perm) {
        vector<int> fact(size(perm) + 1);
        fact[0] = 1;
        for (int i = 0; i + 1 < size(fact); ++i) {
            fact[i + 1] = (static_cast<int64_t>(i + 1) * fact[i]) % MOD;
        }
        int result = 0;
        BIT bit(size(perm));
        for (int i = 0; i < size(perm); ++i) {
            result = (result + (static_cast<int64_t>((((perm[i] - 1) - bit.query((perm[i] - 1) - 1)) % MOD + MOD) % MOD) * fact[(size(perm) - 1) - i] % MOD)) % MOD;
            bit.add(perm[i] - 1, +1);
        }
        return result;
    }

private:
    class BIT {
    public:
        BIT(int n) : bit_(n + 1) {  // 0-indexed
        }
        
        void add(int i, int val) {
            ++i;
            for (; i < size(bit_); i += lower_bit(i)) {
                bit_[i] = (bit_[i] + val) % MOD;
            }
        }

        int query(int i) const {
            ++i;
            int total = 0;
            for (; i > 0; i -= lower_bit(i)) {
                total = (total + bit_[i]) % MOD;
            }
            return total;
        }
    
    private:
        inline int lower_bit(int i) const {
            return i & -i;
        }
        
        vector<int> bit_;
    };
};

// Time:  O(nlogn)
// Space: O(n)
// bit, fenwick tree, combinatorics
class Solution2 {
private:
    static const uint32_t MOD = 1e9 + 7;

public:
    int getPermutationIndex(vector<int>& perm) {
        vector<int> fact = {1, 1};
        const auto& factorial = [&](int n) {
             while (size(fact) <= n) {  // lazy initialization
                fact.emplace_back((static_cast<int64_t>(fact.back()) * size(fact)) % MOD);
            }
            return fact[n];
        };

        int result = 0;
        BIT bit(size(perm));
        for (int i = 0; i < size(perm); ++i) {
            result = (result + (static_cast<int64_t>((((perm[i] - 1) - bit.query((perm[i] - 1) - 1)) % MOD + MOD) % MOD) * factorial((size(perm) - 1) - i) % MOD)) % MOD;
            bit.add(perm[i] - 1, +1);
        }
        return result;
    }

private:
    class BIT {
    public:
        BIT(int n) : bit_(n + 1) {  // 0-indexed
        }
        
        void add(int i, int val) {
            ++i;
            for (; i < size(bit_); i += lower_bit(i)) {
                bit_[i] = (bit_[i] + val) % MOD;
            }
        }

        int query(int i) const {
            ++i;
            int total = 0;
            for (; i > 0; i -= lower_bit(i)) {
                total = (total + bit_[i]) % MOD;
            }
            return total;
        }
    
    private:
        inline int lower_bit(int i) const {
            return i & -i;
        }
        
        vector<int> bit_;
    };
};