201084: [AtCoder]ARC108 E - Random IS

Memory Limit：1024 MB Time Limit：3 S
Judge Style：Text Compare Creator：
Submit：0 Solved：0

Description

Score : $700$ points

Problem Statement

There are $N$ isu - chairs in Japanese - arranged from left to right. The $i$-th chair from the left has the ID number $a_i$. Here, it is guaranteed that $a_i$ are distinct.

Snuke has decided to mark some of the chairs and throw away the rest. Initially, no chair is marked. We call a choice of marked chairs good when the IDs of the marked chairs are monotonically increasing from left to right.

Snuke has decided to do the following procedure to mark chairs:

  1. We say a chair $x$ to be nice if (and only if) the choice of marked chairs is still good when $x$ gets marked. Let $k$ be the current number of nice chairs.
  2. If $k=0$, remove the unmarked chairs and terminate the procedure. Otherwise, choose one from the $k$ nice chairs with equal probability, mark it, and go back to Step 1.

It can be proved that the expected value of the number of chairs that remain at the end of the procedure is a rational number. Let this value be $P/Q$, an irreducible fraction. Additionally, let $M=10^{9}+7$. Then, we can prove that there uniquely exists an integer $R$ between $0$ and $M-1$ (inclusive) such that $P \equiv Q \times R \pmod{M}$, and that value equals $P \times Q^{-1} \pmod{M}$, where $Q^{-1}$ is the modular multiplicative inverse of $Q$. Find $R$.

Constraints

  • All values in input are integers.
  • $1 \leq N \leq 2000$
  • $1 \leq a_i \leq N$
  • $a_i$ are distinct.

Input

Input is given from Standard Input in the following format:

$N$
$a_1$ $a_2$ $\cdots$ $a_N$

Output

Print $R$.

Sample Input 1

3
3 1 2

Sample Output 1

666666673
  • If Chair $1$ (the chair with the ID number $1$) gets marked first, two chairs will remain at the end: Chair $1$ and $2$.
  • If Chair $2$ gets marked first, two chairs will remain at the end: Chair $1$ and $2$.
  • If Chair $3$ gets marked first, one chair will remain at the end: Chair $3$.
  • Since chairs are chosen with equal probability, the expected value of the number of chairs at the end is $\frac{5}{3}$. We have $5 \equiv 3 \times 666666673 \pmod{M}$, so $R=666666673$.

Sample Input 2

30
26 16 28 30 23 11 29 18 22 15 20 13 27 9 21 7 5 25 4 19 8 3 1 24 10 14 17 12 2 6

Sample Output 2

297703424

Input

题意翻译

从左到右排列了 $N$ 张椅子，第 $i$ 张的编号为 $a_i$ （保证 $a_i$ 互不相同）。 $\texttt{Snuke}$ 想要标记一些椅子并把剩下的丢掉，一开始所有椅子都没有被标记。我们称一种标记方案是好的，当且仅当其标号递增。即，若标记的编号为 $i_1<i_2<\cdots<i_k$，则有 $a_{i_1}<a_{i_2}<\cdots<a_{i_k}$。 $\texttt{Snuke}$ 将重复一下操作来标记椅子： 1. 称 $x$ 是不错的当且仅当把 $x$ 加入后标记方案仍是好的，记其数量为 $k$； 2. 若 $k=0$ 结束操作，否则均匀随机选择一个标记并继续操作 $1$； 求最终标记个数的期望。

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