201572: [AtCoder]ARC157 C - YY Square

Problem Statement

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 Ys 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 $998244353$.

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:

$H$ $W$
$S_1$
$S_2$
$\vdots$
$S_H$

Output

Print the sum of the scores over all possible paths, modulo $998244353$.

Sample Input 1

2 2
YY
XY

Sample 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 Ys 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 Ys , so the score is $0^2 = 0$.

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

Sample Input 2

2 2
XY
YY

Sample Output 2

2

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

Sample Input 3

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

Sample Output 3

423787835

Print the sum of the scores modulo $998244353$.