200653: [AtCoder]ARC065 F - Shuffling

Problem Statement

There is a string $S$ of length $N$ consisting of characters 0 and 1. You will perform the following operation for each $i = 1, 2, ..., m$:

• Arbitrarily permute the characters within the substring of $S$ starting at the $l_i$-th character from the left and extending through the $r_i$-th character.

Here, the sequence $l_i$ is non-decreasing.

How many values are possible for $S$ after the $M$ operations, modulo $1000000007(= 10^9+7)$?

Constraints

• $2≦N≦3000$
• $1≦M≦3000$
• $S$ consists of characters 0 and 1.
• The length of $S$ equals $N$.
• $1≦l_i < r_i≦N$
• $l_i ≦ l_{i+1}$

Input

The input is given from Standard Input in the following format:

$N$ $M$
$S$
$l_1$ $r_1$
:
$l_M$ $r_M$

Output

Print the number of the possible values for $S$ after the $M$ operations, modulo $1000000007$.

Sample Input 1

5 2
01001
2 4
3 5

Sample Output 1

6

After the first operation, $S$ can be one of the following three: 01001, 00101 and 00011.

After the second operation, $S$ can be one of the following six: 01100, 01010, 01001, 00011, 00101 and 00110.

Sample Input 2

9 3
110111110
1 4
4 6
6 9

Sample Output 2

26

Sample Input 3

11 6
00101000110
2 4
2 3
4 7
5 6
6 10
10 11

Sample Output 3

143