The package for this problem was not updated by the problem writer or Codeforces administration after we've upgraded the judging servers. To adjust the time limit constraint, a solution execution time will be multiplied by 2. For example, if your solution works for 400 ms on judging servers, then the value 800 ms will be displayed and used to determine the verdict.

Codeforces Round 157 (Div. 1) |
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Finished |

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dp

math

probabilities

*2600

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D. Little Elephant and Broken Sorting

time limit per test

2 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputThe Little Elephant loves permutations of integers from 1 to *n* very much. But most of all he loves sorting them. To sort a permutation, the Little Elephant repeatedly swaps some elements. As a result, he must receive a permutation 1, 2, 3, ..., *n*.

This time the Little Elephant has permutation *p*_{1}, *p*_{2}, ..., *p*_{n}. Its sorting program needs to make exactly *m* moves, during the *i*-th move it swaps elements that are at that moment located at the *a*_{i}-th and the *b*_{i}-th positions. But the Little Elephant's sorting program happened to break down and now on every step it can equiprobably either do nothing or swap the required elements.

Now the Little Elephant doesn't even hope that the program will sort the permutation, but he still wonders: if he runs the program and gets some permutation, how much will the result of sorting resemble the sorted one? For that help the Little Elephant find the mathematical expectation of the number of permutation inversions after all moves of the program are completed.

We'll call a pair of integers *i*, *j* (1 ≤ *i* < *j* ≤ *n*) an inversion in permutatuon *p*_{1}, *p*_{2}, ..., *p*_{n}, if the following inequality holds: *p*_{i} > *p*_{j}.

Input

The first line contains two integers *n* and *m* (1 ≤ *n*, *m* ≤ 1000, *n* > 1) — the permutation size and the number of moves. The second line contains *n* distinct integers, not exceeding *n* — the initial permutation. Next *m* lines each contain two integers: the *i*-th line contains integers *a*_{i} and *b*_{i} (1 ≤ *a*_{i}, *b*_{i} ≤ *n*, *a*_{i} ≠ *b*_{i}) — the positions of elements that were changed during the *i*-th move.

Output

In the only line print a single real number — the answer to the problem. The answer will be considered correct if its relative or absolute error does not exceed 10^{ - 6}.

Examples

Input

2 1

1 2

1 2

Output

0.500000000

Input

4 3

1 3 2 4

1 2

2 3

1 4

Output

3.000000000

Codeforces (c) Copyright 2010-2024 Mike Mirzayanov

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