docker_space/frontier_cs_8/count_vals.cpp

#include <bits/stdc++.h>
using namespace std;

static const long long P = 998244353;

int main() {
    // For d=10, enumerate all d!=3628800 configs and count distinct T values
    for (int d = 3; d <= 12; d++) {
        unordered_set<long long> vals;
        vector<int> gy(d, 0);

        while (true) {
            long long Sv[35];
            Sv[0] = 0;
            for (int j = 0; j < d; j++) {
                long long c = ((Sv[j] - Sv[gy[j]] + (j - gy[j]) + 1) % P + P) % P;
                Sv[j + 1] = (Sv[j] + c) % P;
            }
            long long T = (Sv[d] + d + 1) % P;
            vals.insert(T);

            int pos = d - 1;
            while (pos >= 0) {
                gy[pos]++;
                if (gy[pos] <= pos) break;
                gy[pos] = 0;
                pos--;
            }
            if (pos < 0) break;
        }

        long long total = 1;
        for (int j = 1; j <= d; j++) total *= j;
        // But gy[0] is always 0, so actual configs = d!/1 = d!... wait gy[0] must be 0
        // Actually gy[j] ranges from 0 to j, so there are (j+1) choices for gy[j]
        // Total = product(j+1, j=0..d-1) = product(1,2,...,d) = d!
        // Wait no: gy[0] can only be 0 (0..0), gy[1] can be 0 or 1, etc.
        // So total = 1 * 2 * 3 * ... * d = d!
        cout << "d=" << d << ": " << vals.size() << " distinct values out of " << total << " configs" << endl;
    }
    return 0;
}