YY Square

AtCoder

IDarc157_c

Time3000ms

Memory256MB

Difficulty—

There is a grid with $H$ rows and $W$ columns where each square has one of the characters `X` and `Y` written on it. Let $(i, j)$ denote the square at the $i$\-th row from the top and $j$\-th column from the left. The characters on the grid are given as $H$ strings $S_1, S_2, \dots, S_H$: the $j$\-th character of $S_i$ is the character on square $(i, j)$. For a path $P$ from square $(1, 1)$ to square $(H, W)$ obtained by repeatedly moving down or right to the adjacent square, the **score** is defined as follows. * Let $\mathrm{str}(P)$ be the string of length $(H + W - 1)$ obtained by concatenating the characters on the squares visited by $P$ in the order they are visited. * The score of $P$ is the **square of the number** of pairs of consecutive `Y`s in $\mathrm{str}(P)$. There are $\displaystyle\binom{H + W - 2}{H - 1}$ such paths. Find the sum of the scores over all those paths, modulo $998244353$. What is $\binom{N}{K}$? $\displaystyle\binom{N}{K}$ is the binomial coefficient representing the number of ways to choose $K$ from $N$ distinct elements. ## Constraints * $1 \leq H \leq 2000$ * $1 \leq W \leq 2000$ * $S_i \ (1 \leq i \leq H)$ is a string of length $W$ consisting of `X` and `Y`. ## Input The input is given from Standard Input in the following format: $H$ $W$ $S_1$ $S_2$ $\vdots$ $S_H$ [samples]

Samples

Input #1

2 2
YY
XY

Output #1

4

There are two possible paths $P$: $(1, 1) \to (1, 2) \to (2, 2)$ and $(1, 1) \to (2, 1) \to (2, 2)$.

*   For $(1, 1) \to (1, 2) \to (2, 2)$, we have $\mathrm{str}(P) = {}$`YYY`, with two pairs of consecutive `Y`s at positions $1, 2$ and $2, 3$, so the score is $2^2 = 4$.
*   For $(1, 1) \to (2, 1) \to (2, 2)$, we have $\mathrm{str}(P) = {}$`YXY`, with no pairs of consecutive `Y`s , so the score is $0^2 = 0$.

Thus, the sought sum is $4 + 0 = 4$.

Input #2

2 2
XY
YY

Output #2

2

For either of the two possible paths $P$, we have $\mathrm{str}(P) = {}$`XYY`, for a score of $1^2 = 1$.

Input #3

10 20
YYYYYYYYYYYYYYYYYYYY
YYYYYYYYYYYYYYYYYYYY
YYYYYYYYYYYYYYYYYYYY
YYYYYYYYYYYYYYYYYYYY
YYYYYYYYYYYYYYYYYYYY
YYYYYYYYYYYYYYYYYYYY
YYYYYYYYYYYYYYYYYYYY
YYYYYYYYYYYYYYYYYYYY
YYYYYYYYYYYYYYYYYYYY
YYYYYYYYYYYYYYYYYYYY

Output #3

423787835

Print the sum of the scores modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "YY Square",
    "description": {
      "content": "There is a grid with $H$ rows and $W$ columns where each square has one of the characters `X` and `Y` written on it. Let $(i, j)$ denote the square at the $i$\\-th row from the top and $j$\\-th column f",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc157_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a grid with $H$ rows and $W$ columns where each square has one of the characters `X` and `Y` written on it. Let $(i, j)$ denote the square at the $i$\\-th row from the top and $j$\\-th column f...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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