Problem:
Score : 1200 points

Problem Statement

One day Mr. Takahashi picked up a dictionary containing all of the N! permutations of integers 1 through N.
The dictionary has N! pages, and page i (1 ≤ i ≤ N!) contains the i-th permutation in the lexicographical order.

Mr. Takahashi wanted to look up a certain permutation of length N in this dictionary, but he forgot some part of it.

His memory of the permutation is described by a sequence P_1, P_2, ..., P_N.
If P_i = 0, it means that he forgot the i-th element of the permutation; otherwise, it means that he remembered the i-th element of the permutation and it is P_i.

He decided to look up all the possible permutations in the dictionary.
Compute the sum of the page numbers of the pages he has to check, modulo 10^9 + 7.

Constraints

1 ≤ N ≤ 500000

0 ≤ P_i ≤ N

P_i ≠ P_j if i ≠ j (1 ≤ i, j ≤ N), P_i ≠ 0 and P_j ≠ 0.

Partial Score

In test cases worth 500 points, 1 ≤ N ≤ 3000.

Input

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

N
P_1 P_2 ... P_N

Output

Print the sum of the page numbers of the pages he has to check, as modulo 10^9 + 7.

Sample Input 1

4
0 2 3 0

Sample Output 1

23

The possible permutations are [1, 2, 3, 4] and [4, 2, 3, 1].
Since the former is on page 1 and the latter is on page 22, the answer is 23.

Sample Input 2

3
0 0 0

Sample Output 2

21

Since all permutations of length 3 are possible, the answer is 1 + 2 + 3 + 4 + 5 + 6 = 21.

Sample Input 3

5
1 2 3 5 4

Sample Output 3

2

Mr. Takahashi remembered all the elements of the permutation.

Sample Input 4

1
0

Sample Output 4

1

The dictionary consists of one page.

Sample Input 5

10
0 3 0 0 1 0 4 0 0 0

Sample Output 5

953330050

Solution in Haskell (GHC 7.10.3) (judge verdict: Wrong Answer)

import Data.List

taka :: [Int] -> Int
taka xs = mod (sum $ map (directionaryOrder) (p5 xs)) (7 + 10 ^ 9)

p5 :: [Int] -> [[Int]]
p5 xs = iter xs (kuwa xs)
    where
        iter [] _ = [[]]
        iter (x:xs) ys
            | x == 0 = do
                x <- ys
                map (x:) (iter xs ( delete x ys ))
            | x /= 0 = map (x:) (iter xs ys)

kuwa :: [Int] -> [Int]
kuwa xs = filter (\x -> not (elem x xs)) [1..(length xs)]

directionaryOrder :: [Int] -> Int
directionaryOrder xs = iter xs (sort xs) 1
    where
        iter [] _ acc = mod acc (7 + 10 ^ 9)
        iter (x:xs) all acc = iter (xs) (delete x all) (acc + (order all x) * factorial (length xs))

order :: [Int] -> Int -> Int
order xs y = iter xs y 0
    where
        iter (x:xs) y acc
            | x == y = acc
            | x /= y = iter xs y (acc + 1)

factorial :: Integral a => a -> a
factorial n = iter n 1
    where
        iter 0 acc = acc
        iter n acc = iter (n-1) (n*acc)

main = do
    n' <- getLine
    p' <- getLine
    print $ taka $ map (\x -> (read x :: Int)) (words p')