A16140. Expected Destruction

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

Each second you do the following steps:

1. Pick an element $x$ in $S$ uniformly at random.
2. Remove $x$ from $S$ .
3. 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 $1\,000\,000\,007$ .

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

1. $[1, 3]$ (50% chance) $\to$ $[1]$ $\to$ $[2]$ $\to$ $[3]$ $\to$ $[]$
2. $[1, 3]$ (50% chance) $\to$ $[2, 3]$ (50% chance) $\to$ $[2]$ $\to$ $[3]$ $\to$ $[]$
3. $[1, 3]$ (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 $750000009 \cdot 4 \equiv 15 \pmod{1\,000\,000\,007}$ .