(Analysis by Daniel Zhang, Benjamin Qi)

The problem is asking for the number of arrangements obtainable if we are allowed to transpose adjacent elements if they differ by exactly one.

Another way of phrasing the problem is as follows: Consider a graph $G$ with vertices labeled $1\ldots N$ and a directed edge from $i$ to $j$ if $i<j$ and $|h_i-h_j|\neq 1$. Count the number of topological sorts of $G$.

Subtask 2 ($1\le h_i\le 3$):

The answer is just $\binom{N}{(\#\text{ of }2\text{s})}$.

Full Solution 1:

First, let's consider the problem of determining if a fixed arrangement is obtainable. If we ignore the constraint on which elements we can transpose, we can determine where each element will end up and use insertion sort. Namely, for each element in order from left to right, move it left some number of times. In fact, this still works because insertion sort will only swap elements that must be swapped in any sequence of transpositions.

Thus, we can build all obtainable arrangements by inserting elements one by one into partial arrangements.

For the third subtask ($|h_i-i|\le 1$), we can apply this insertion sort idea and notice that we only need to keep track of the last three elements of the prefix, because it is impossible for a haystack to move left more than three times.

Since we only care about the number of arrangements, most arrangements are indistinguishable. If we are about to insert $x$, we only really care about the longest suffix consisting of only $x-1$ and $x+1$. This information can also be easily updated after each insertion. These values will correspond to our states in our DP, along with the length of the partial arrangement.

These values are zero for almost all values of $x$. We could equivalently store the last element ($a$), the length of the longest suffix containing only $a$ and $a+2$ ($\ell_1$), and the length of the longest suffix containing only $a$ and $a-2$ ($\ell_2$).

It turns out after inserting any prefix of elements of length $n$, only $O(n)$ states are obtainable. Specifically, $a\in [x-1,x+1]$, and if we fix $a$, we can show that either

$$(\ell_1,\ell_2)\in \{(i,\min(i,\ell_a)): 1\le i\le n\}$$

or

$$(\ell_1,\ell_2)\in \{(\min(i,\ell_a),i): 1\le i\le n\},$$

where $\ell_a$ is the number of occurrences of $a$ in the longest suffix of the prefix containing no elements outside of $a-1$, $a$, and $a+1$. In fact, we can actually show something stronger: at most $2n$ states are attainable, since $a$ can only take on two values.

As there are $\mathcal O(n)$ states and $\mathcal O(n)$ transitions from each, this automatically yields a solution that runs in $\mathcal O(N^3)$ (possibly with additional log factors depending on the implementation). To speed this up, we can do the transitions using prefix sums, yielding a $O(N^2)$ solution. Slower implementations (ex. with an additional factor of $\log N$ from $\texttt{std::map}$ might not receive full credit).

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

const int MOD = 1e9+7;

#define f first
#define s second

struct mi {
int v;
mi(int _v): v(_v) {}
mi& operator+=(const mi& o) {
v += o.v;
if (v >= MOD) v -= MOD;
return *this;
}
};

struct SuffixInfo {
int max_1 = INT_MAX, max_2 = INT_MAX;
vector<mi> dp;
void add(pair<int,int> p, mi t) {
if (p.f < p.s) {
if (max_1 != INT_MAX) assert(max_1 == p.f);
else max_1 = p.f;
}
if (p.f > p.s) {
if (max_2 != INT_MAX) assert(max_2 == p.s);
else max_2 = p.s;
}
const int idx = max(p.f,p.s);
while ((int)size(dp) <= idx) dp.push_back(0);
dp.at(idx) += t;
}
// returns {ell_1, ell_2}
pair<int,int> query(int x) { return {min(max_1,x),min(max_2,x)}; }
void partial_sum() {
for (int i = (int)size(dp)-2; i > 0; --i)
dp[i] += dp[i+1];
}
};

int prev_x;
array<SuffixInfo,3> info;

array<SuffixInfo,3> new_info;
for (int offset = 0; offset < 3; ++offset) {
const int last_h = prev_x+offset-1;
for (int i = 0; i < (int)size(info[offset].dp); ++i) {
auto [ways1, ways2] = info[offset].query(i);
auto query = [&](int y) {
if (y == last_h-1) return ways1;
if (y == last_h+1) return ways2;
return 0;
};
{ // case 1: x goes in front
}
if (x == last_h+1) {
}
if (x == last_h-1) {
}
}
}
new_info.at(0).partial_sum();
new_info.at(2).partial_sum();
prev_x = x;
swap(info,new_info);
}

int main() {
int T; cin >> T;
while (T--) {
int N; cin >> N;
for (int i = 0; i < 3; ++i)
info[i] = SuffixInfo();
cin >> prev_x;
for (int i = 1; i < N; ++i) {
int h; cin >> h;
}
mi ans = 0;
for (const SuffixInfo& a: info)
for (const mi& b: a.dp) ans += b;
cout << ans.v << "\n";
}
}


Solution 2: The key observation is that the relative order of the haybales with even heights never changes, and that the same holds for the odd heights. In other words, if $i<j$ and $h_i$ and $h_j$ have the same parity then there is definitely an edge $i\to j$ in $G$. Thinking about either subtasks 2 or 4 might have made this easier to see.

(Corollary: The answer is bounded above by $\binom{N}{\lfloor N/2\rfloor}$.)

The idea is to determine the elements of the final configuration from left to right. Let $\texttt{dp}[i][j]$ denote the number of ways to determine the first $i+j$ elements of the final configuration such that it contains the first $i$ odd heights and the first $j$ even heights. Given $\texttt{dp}[i][j]$, we can advance to $\texttt{dp}[i+1][j]$ if we can choose the next element of the final configuration to be even, or $\texttt{dp}[i][j+1]$ if can choose the next element of the final configuration to be odd?

When is it okay to advance from $\texttt{dp}[i][j]$ to $\texttt{dp}[i+1][j]$? It turns out that this is allowed when both $(i,j)$ and $(i+1,j)$ are feasible:

Let the indices of the odd heights be $h_{o_0}, h_{o_1},\ldots, h_{o_{\texttt{odd}-1}}$ and the indices of the even heights be $h_{e_0}, h_{e_1},\ldots, h_{e_{\texttt{even}-1}}$. For all pairs $(i,j)$ satisfying $0\le i\le \texttt{odd}$ and $0\le j\le \texttt{even}$, say that $(i,j)$ is feasible if there do not exist $(i',j')$ satisfying one of the following two sets of conditions:

• $0\le i'<i$ and $j\le j'< \texttt{even}$ such that $o_{i'}>e_{j'}$ and $|h_{o_{i'}}-h_{e_{j'}}|>1$
• $i\le i'<\texttt{odd}$ and $0\le j'<j$ such that $o_{i'}<e_{j'}$ and $|h_{o_{i'}}-h_{e_{j'}}|>1$

So it turns out that all we need to do is count the number of paths from $(0,0)$ to $(\texttt{odd},\texttt{even})$ that only pass through feasible pairs. This can be done in $\mathcal{O}(\texttt{odd}\cdot \texttt{even})\le \mathcal{O}(N^2)$ time.

import java.io.BufferedReader;
import java.io.IOException;
import java.util.StringTokenizer;

public class Haybales {
public static final long MOD = 1000000007;

public static void main(String[] args) throws IOException {
for (int tc = 0; tc < tcs; ++tc) {
int[] heights = new int[n];
int[] position = new int[n];
int odd = 0;
int even = 0;
for (int j = 0; j < n; j++) {
heights[j] = Integer.parseInt(tokenizer.nextToken());
if (heights[j] % 2 == 1) {
odd++;
position[j] = odd;
} else {
even++;
position[j] = even;
}
}
int[][] after = {new int[even + 1], new int[odd + 1]};
for (int j = 0; j < n; j++) {
for (int k = j + 1; k < n; k++) {
if (heights[j] % 2 != heights[k] % 2 && Math.abs(heights[j] - heights[k]) >= 2) {
after[heights[k] % 2][position[k]] = Math.max(after[heights[k] % 2][position[k]], position[j]);
}
}
}
long[][] dp = new long[odd + 1][even + 1];
dp = 1;
for (int j = 0; j <= odd; j++) {
for (int k = 0; k <= even; k++) {
if (j > 0 && after[j] <= k) {
dp[j][k] += dp[j - 1][k];
}
if (k > 0 && after[k] <= j) {
dp[j][k] += dp[j][k - 1];
}
dp[j][k] %= MOD;
}
}
System.out.println(dp[odd][even]);
}
}
}


Bonus: It is possible to solve this problem in $\mathcal O(N\log N)$ time with FFT.