Expected Destruction
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 $z \cdot b \equiv a \pmod{P}$.
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 a single integer — the expected number of seconds until $S$ is empty, modulo $1\,000\,000\,007$.
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 $1\,000\,000\,007$.
2 3
1 3
5 10
1 2 3 4 5
5 10
2 3 6 8 9
1 100
1
750000009
300277731
695648216
100
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 $750000009 \cdot 4 \equiv 15 \pmod{1\,000\,000\,007}$.