9248: ABC293 —— C - Make Takahashi Happy
[Creator : ]
Description
### Problem Statement
There is a grid with $H$ horizontal rows and $W$ vertical columns. For two integers $i$ and $j$ such that $1 \leq i \leq H$ and $1 \leq j \leq W$, the square at the $i$-th row from the top and $j$-th column from the left (which we denote by $(i, j)$) has an integer $A_{i, j}$ written on it.
Takahashi is currently at $(1,1)$. From now on, he repeats moving to an adjacent square to the right of or below his current square until he reaches $(H, W)$. When he makes a move, he is not allowed to go outside the grid.
Takahashi will be happy if the integers written on the squares he visits (including initial $(1, 1)$ and final $(H, W)$) are distinct. Find the number of his possible paths that make him happy.
There is a grid with $H$ horizontal rows and $W$ vertical columns. For two integers $i$ and $j$ such that $1 \leq i \leq H$ and $1 \leq j \leq W$, the square at the $i$-th row from the top and $j$-th column from the left (which we denote by $(i, j)$) has an integer $A_{i, j}$ written on it.
Takahashi is currently at $(1,1)$. From now on, he repeats moving to an adjacent square to the right of or below his current square until he reaches $(H, W)$. When he makes a move, he is not allowed to go outside the grid.
Takahashi will be happy if the integers written on the squares he visits (including initial $(1, 1)$ and final $(H, W)$) are distinct. Find the number of his possible paths that make him happy.
Input
### Input
The input is given from Standard Input in the following format:
```
$H$ $W$
$A_{1, 1}$ $A_{1, 2}$ $\ldots$ $A_{1, W}$
$A_{2, 1}$ $A_{2, 2}$ $\ldots$ $A_{2, W}$
$\vdots$
$A_{H, 1}$ $A_{H, 2}$ $\ldots$ $A_{H, W}$
```
The input is given from Standard Input in the following format:
```
$H$ $W$
$A_{1, 1}$ $A_{1, 2}$ $\ldots$ $A_{1, W}$
$A_{2, 1}$ $A_{2, 2}$ $\ldots$ $A_{2, W}$
$\vdots$
$A_{H, 1}$ $A_{H, 2}$ $\ldots$ $A_{H, W}$
```
Output
### Output
Print the answer.
Print the answer.
Constraints
### Constraints
- $2 \leq H, W \leq 10$
- $1 \leq A_{i, j} \leq 10^9$
- All values in the input are integers.
- $2 \leq H, W \leq 10$
- $1 \leq A_{i, j} \leq 10^9$
- All values in the input are integers.
Sample 1 Input
3 3
3 2 2
2 1 3
1 5 4
Sample 1 Output
3
There are six possible paths:
- (1,1)→(1,2)→(1,3)→(2,3)→(3,3): the integers written on the squares he visits are 3,2,2,3,4, so he will not be happy.
- (1,1)→(1,2)→(2,2)→(2,3)→(3,3): the integers written on the squares he visits are 3,2,1,3,4, so he will not be happy.
- (1,1)→(1,2)→(2,2)→(3,2)→(3,3): the integers written on the squares he visits are 3,2,1,5,4, so he will be happy.
- (1,1)→(2,1)→(2,2)→(2,3)→(3,3): the integers written on the squares he visits are 3,2,1,3,4, so he will not be happy.
- (1,1)→(2,1)→(2,2)→(3,2)→(3,3): the integers written on the squares he visits are 3,2,1,5,4, so he will be happy.
- (1,1)→(2,1)→(3,1)→(3,2)→(3,3): the integers written on the squares he visits are 3,2,1,5,4, so he will be happy.
Sample 2 Input
10 10
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Sample 2 Output
48620
In this example, every possible path makes him happy.