101834: [AtCoder]ABC183 E - Queen on Grid

Problem Statement

We have a grid with $H$ horizontal rows and $W$ vertical columns of squares. Square $(i,j)$, which is at the $i$-th row from the top and $j$-th column from the left, is wall if $S_{ij}$ is # and road if $S_{ij}$ is ..

There is a queen, the chess piece, at Square $(1, 1)$. In one move, it can move any number of squares to the right, downwards, or diagonally to the lower right to a road square without jumping over wall squares.

In how many ways can the queen travel from Square $(1, 1)$ to Square $(H, W)$? Find the count modulo $(10^9+7)$.

Here, two ways to travel are considered different if and only if there exists $i$ such that the position of the queen after the $i$-th move is different in those two ways.

Constraints

• $2 \leq H,W \leq 2000$
• $S_{ij}$ is # or ..
• $S_{11}$ and $S_{HW}$ are ..

Input

Input is given from Standard Input in the following format:

$H$ $W$
$S_{11}\ldots S_{1W}$
$\vdots$
$S_{H1}\ldots S_{HW}$

Output

Print the number of ways, modulo $(10^9+7)$, in which the queen can travel from Square $(1, 1)$ to Square $(H, W)$.

Sample Input 1

3 3
...
.#.
...

Sample Output 1

10

There are $10$ ways to travel, as follows:

• $(1,1)\to (1,2)\to (1,3)\to (2,3)\to (3,3)$
• $(1,1)\to (1,2)\to (1,3)\to (3,3)$
• $(1,1)\to (1,2)\to (2,3)\to (3,3)$
• $(1,1)\to (1,3)\to (2,3)\to (3,3)$
• $(1,1)\to (1,3)\to (3,3)$
• $(1,1)\to (2,1)\to (3,1)\to (3,2)\to (3,3)$
• $(1,1)\to (2,1)\to (3,1)\to (3,3)$
• $(1,1)\to (2,1)\to (3,2)\to (3,3)$
• $(1,1)\to (3,1)\to (3,2)\to (3,3)$
• $(1,1)\to (3,1)\to (3,3)$

Sample Input 2

4 4
...#
....
..#.
....

Sample Output 2

84

From $(1,1)$, the queen can move to $(1,2)$, $(1,3)$, $(2,1)$, $(2,2)$, $(3,1)$, or $(4,1)$.

One possible path to $(4,4)$ is $(1,1)\to (3,1)\to (3,2)\to (4,3)\to (4,4)$.

Sample Input 3

8 10
..........
..........
..........
..........
..........
..........
..........
..........

Sample Output 3

13701937

Find the count modulo $(10^9+7)$.