## USACO 2022 January Contest, Platinum

## Problem 2. Counting Haybales

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As usual, Bessie the cow is causing trouble in Farmer John's barn. FJ has $N$
($1\leq N \leq 5000$) stacks of haybales. For each $i\in [1,N]$, the $i$th stack
has $h_i$ ($1\le h_i\le 10^9$) haybales. Bessie does not want any haybales to
fall, so the only operation she can perform is as follows:

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- If two adjacent stacks' heights differ by exactly one, she can move the top haybale of the taller stack to the shorter stack.

How many configurations are obtainable after performing the above operation finitely many times, modulo $10^9+7$? Two configurations are considered the same if, for all $i$, the $i$th stack has the same number of haybales in both.

#### INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $T$ ($1\le T\le 10$), the number of independent test cases, all of which must be solved to solve one input correctly.Each test case consists of $N$, and then a sequence of $N$ heights. It is guaranteed that the sum of $N$ over all test cases does not exceed $5000$.

#### OUTPUT FORMAT (print output to the terminal / stdout):

Please output $T$ lines, one for each test case.#### SAMPLE INPUT:

7 4 2 2 2 3 4 3 3 1 2 4 5 3 4 2 6 3 3 1 1 2 2 6 1 3 3 4 1 2 6 4 1 2 3 5 4 10 1 5 6 6 6 4 2 3 2 5

#### SAMPLE OUTPUT:

4 4 5 15 9 8 19

For the first test case, the four possible configurations are:

$$(2,2,2,3), (2,2,3,2), (2,3,2,2), (3,2,2,2).$$

For the second test case, the four possible configurations are:

$$(2,3,3,1),(3,2,3,1),(3,3,2,1), (3,3,1,2).$$

#### SCORING:

- Inputs 1-3 satisfy $N\le 10$.
- Input 4 satisfies $1\le h_i\le 3$ for all $i$.
- Inputs 5-7 satisfy $|h_i-i|\le 1$ for all $i$.
- Inputs 8-10 satisfy $1\le h_i\le 4$ for all $i$ and $N\le 100$.
- Inputs 11-13 satisfy $N\le 100$.
- Inputs 14-17 satisfy $N\le 1000$.
- Inputs 18-21 satisfy no additional constraints.

Problem credits: Daniel Zhang