Expected Destruction

1000ms

262144K

Description:

You have a set $$$S$$$ of $$$n$$$ distinct integers between $$$1$$$ and $$$m$$$.

Each second you do the following steps:

Pick an element $$$x$$$ in $$$S$$$ uniformly at random.
Remove $$$x$$$ from $$$S$$$.
If $$$x+1 \leq m$$$ and $$$x+1$$$ is not in $$$S$$$, add $$$x+1$$$ to $$$S$$$.

What is the expected number of seconds until $$$S$$$ is empty?

Output the answer modulo $$$1\,000\,000\,007$$$.

Formally, let $$$P = 1\,000\,000\,007$$$. It can be shown that the answer can be expressed as an irreducible fraction $$$\frac{a}{b}$$$, where $$$a$$$ and $$$b$$$ are integers and $$$b \not \equiv 0 \pmod{P}$$$. Output the integer equal to $$$a \cdot b^{-1} \bmod P$$$. In other words, output an integer $$$z$$$ such that $$$0 \le z < P$$$ and $$$z \cdot b \equiv a \pmod{P}$$$.

Input:

The first line contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n \leq m \leq 500$$$) — the number of elements in the set $$$S$$$ and the upper bound on the value of the elements in $$$S$$$.

The second line contains $$$n$$$ integers $$$S_1,\,S_2,\,\dots,\,S_n$$$ ($$$1 \leq S_1 < S_2 < \ldots < S_n \leq m$$$) — the elements of the set $$$S$$$.

Output:

Output a single integer — the expected number of seconds until $$$S$$$ is empty, modulo $$$1\,000\,000\,007$$$.

Sample Input:

2 3
1 3

Sample Output:

750000009

Sample Input:

5 10
1 2 3 4 5

Sample Output:

300277731

Sample Input:

5 10
2 3 6 8 9

Sample Output:

695648216

Sample Input:

1 100
1

Sample Output:

Note:

For test 1, here is a list of all the possible scenarios and their probabilities:

$$$[1, 3]$$$ (50% chance) $$$\to$$$ $$$[1]$$$ $$$\to$$$ $$$[2]$$$ $$$\to$$$ $$$[3]$$$ $$$\to$$$ $$$[]$$$
$$$[1, 3]$$$ (50% chance) $$$\to$$$ $$$[2, 3]$$$ (50% chance) $$$\to$$$ $$$[2]$$$ $$$\to$$$ $$$[3]$$$ $$$\to$$$ $$$[]$$$
$$$[1, 3]$$$ (50% chance) $$$\to$$$ $$$[2, 3]$$$ (50% chance) $$$\to$$$ $$$[3]$$$ $$$\to$$$ $$$[]$$$

Adding them up, we get $$$\frac{1}{2}\cdot 4 + \frac{1}{4} \cdot 4 + \frac{1}{4} \cdot 3 = \frac{15}{4}$$$. We see that $$$750000009 \cdot 4 \equiv 15 \pmod{1\,000\,000\,007}$$$.

Informação

Provedor Codeforces

Código CF1854C