Make Takahashi Happy

AtCoder

IDabc293_c

Time2000ms

Memory256MB

Difficulty—

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. ## Constraints * $2 \leq H, W \leq 10$ * $1 \leq A_{i, j} \leq 10^9$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $H$ $W$ $A_{1, 1}$ $A_{1, 2}$ $\ldots$ $A_{1, W}$ $A_{2, 1}$ $A_{2, 2}$ $\ldots$ $A_{2, W}$ $\vdots$ $A_{H, 1}$ $A_{H, 2}$ $\ldots$ $A_{H, W}$ [samples]

Samples

Input #1

Output #1

3

There are six possible paths:

*   $(1, 1) \rightarrow (1, 2) \rightarrow (1, 3) \rightarrow (2, 3) \rightarrow (3, 3)$: the integers written on the squares he visits are $3, 2, 2, 3, 4$, so he **will not** be happy.
*   $(1, 1) \rightarrow (1, 2) \rightarrow (2, 2) \rightarrow (2, 3) \rightarrow (3, 3)$: the integers written on the squares he visits are $3, 2, 1, 3, 4$, so he **will not** be happy.
*   $(1, 1) \rightarrow (1, 2) \rightarrow (2, 2) \rightarrow (3, 2) \rightarrow (3, 3)$: the integers written on the squares he visits are $3, 2, 1, 5, 4$, so he **will** be happy.
*   $(1, 1) \rightarrow (2, 1) \rightarrow (2, 2) \rightarrow (2, 3) \rightarrow (3, 3)$: the integers written on the squares he visits are $3, 2, 1, 3, 4$, so he **will not** be happy.
*   $(1, 1) \rightarrow (2, 1) \rightarrow (2, 2) \rightarrow (3, 2) \rightarrow (3, 3)$: the integers written on the squares he visits are $3, 2, 1, 5, 4$, so he **will** be happy.
*   $(1, 1) \rightarrow (2, 1) \rightarrow (3, 1) \rightarrow (3, 2) \rightarrow (3, 3)$: the integers written on the squares he visits are $3, 2, 1, 5, 4$, so he **will** be happy.

Thus, the third, fifth, and sixth paths described above make him happy.

Input #2

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

Output #2

48620

In this example, every possible path makes him happy.

API Response (JSON)

{
  "problem": {
    "name": "Make Takahashi Happy",
    "description": {
      "content": "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 $j$\\-t",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc293_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 $j$\\-t...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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