/**
* Quite simple, though a little bit tedious?
*
* Time Complexity: O(n * log(n)) (Full solution)
* Implementation 1 (Math, implementation)
*/
#include <bits/stdc++.h>
#include "gondola.h"
typedef long long ll;
typedef std::vector<int> vec;
const ll MOD = 1e9 + 9;
vec cyc_shift(int n, int input_seq[]) {
int original = -1, original_pos = -1;
for (int k = 0; k < n && original == -1; k++) {
if (input_seq[k] <= n)
original = input_seq[k], original_pos = k;
}
vec shift(n);
int delta = (original != -1 ? original_pos + 1 - original : 0);
for (int k = 0; k < n; k++)
shift[k] = input_seq[(k + delta + n) % n];
return shift;
}
int valid(int n, int input_seq[]) {
vec shift = cyc_shift(n, input_seq);
bool ok = true;
std::set<int> appeared;
for (int k = 0; k < n && ok; k++) {
if (shift[k] <= n) {
ok &= (shift[k] == k + 1);
} else {
ok &= (appeared.find(shift[k]) == appeared.end());
appeared.emplace(shift[k]);
}
}
return ok;
}
int replacement(int n, int gondola_seq[], int replacement_seq[]) {
assert(valid(n, gondola_seq));
vec shift = cyc_shift(n, gondola_seq);
int repl_len = 0;
vec count(n + 1, 0);
memset(replacement_seq, -1, sizeof(int) * 250001);
for (int k = 0; k < n; k++) {
if (shift[k] > n) {
repl_len = std::max(repl_len, shift[k] - n);
replacement_seq[shift[k] - n - 1] = k + 1;
}
}
for (int k = repl_len - 1, last = -1; k >= 0; k--) {
if (replacement_seq[k] == -1) {
replacement_seq[k] = replacement_seq[last];
replacement_seq[last] = n + k + 1;
}
last = k;
}
return repl_len;
}
int countReplacement(int n, int input_seq[]) {
if (!valid(n, input_seq))
return 0;
vec shift = cyc_shift(n, input_seq);
int max_val = 0;
vec top;
for (int k = 0; k < n; k++) {
if (shift[k] > n)
top.push_back(shift[k]);
max_val = std::max(max_val, shift[k]);
}
std::sort(top.rbegin(), top.rend());
int m = top.size();
ll comb = 1;
for (int t = n + 1; t <= max_val; t++) {
int pt = -1;
for (int step = m / 2 + 1; step >= 1; step /= 2) {
while (pt + step < m && top[pt + step] >= t)
pt += step;
}
if (pt != -1 && top[pt] == t)
continue;
comb *= ll(pt + 1), comb %= MOD;
}
bool fixed_cyc = false;
for (int k = 0; k < n && !fixed_cyc; k++)
fixed_cyc |= (shift[k] == k + 1);
if (!fixed_cyc)
comb *= ll(n), comb %= MOD;
return int(comb);
}
