201195: [AtCoder]ARC119 F - AtCoder Express 3

Problem Statement

There are $N+1$ stations along the AtCoder Line, numbered $0$ through $N$. Formerly, it only had Local Trains, which run between Stations $i$ and $i + 1$ in one minute in either direction for each $i$ $(0 \leq i \leq N-1)$.

One day, however, the railway company broke into two groups, called Ko-soku (light speed) and Jun-kyu (semi express). They decided that each station other than Stations $0$ and $N$ should be administered by one of the groups. The group that administers Station $j$ $(1 \leq j \leq N-1)$ is represented by a character $c_j$: A means that Ko-soku administers the station, B means Jun-kyu administers the station, and ? means it is undecided. Since Stations $0$ and $N$ are so important, both groups will administer them.

Now, Ko-soku and Jun-kyu have decided to run two new types of trains, in addition to the local trains.

Ko-soku Trains: Let Stations $a_0, a_1, \dots, a_s$ be the stations administered by Ko-soku in ascending order. These trains run between Stations $a_k$ and $a_{k+1}$ in one minute in either direction for each $k$.

Jun-kyu Trains: Let Stations $b_0, b_1, \dots, b_t$ be the stations administered by Jun-kyu in ascending order. These trains run between Stations $b_k$ and $b_{k+1}$ in one minute in either direction for each $k$.

There are $2^q$ ways in which these trains run, where $q$ is the number of ?s. Among them, how many enables us to go from Station $0$ to Station $N$ in no more than $K$ minutes' ride? Find this count modulo $(10^9+7)$.

Constraints

• $2 \leq N \leq 4000$
• $1 \leq K \leq \frac{N+1}{2}$
• $N$ and $K$ are integers.
• Each of $c_1, c_2, \dots, c_{N-1}$ is A, B, or ?.

Input

Input is given from Standard Input in the following format:

$N$ $K$
$c_1$$c_2$$\cdots$$c_{N-1}$

Output

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

Sample Input 1

8 3
A??AB?B

Sample Output 1

4

Here, there are $2^3 = 8$ possible ways in which the trains run. Among them, the following four enable us to go from Station $0$ to Station $8$ in no more than $3$ minutes' ride.

• If Ko-soku administers Stations $2, 3, 6$, we can go Station $0 \rightarrow 5 \rightarrow 7 \rightarrow 8$, as shown at #1 in the figure below.
• If Ko-soku administers Stations $2, 3$ and Jun-kyu administers Station $6$, we can go Station $0 \rightarrow 5 \rightarrow 4 \rightarrow 8$, as shown at #2 in the figure below.
• If Ko-soku administers Station $2$ and Jun-kyu administers Stations $3, 6$, we can go Station $0 \rightarrow 3 \rightarrow 4 \rightarrow 8$, as shown at #4 in the figure below.
• If Jun-kyu administers Stations $2, 3, 6$, we can go Station $0 \rightarrow 1 \rightarrow 4 \rightarrow 8$, as shown at #8 in the figure below.

Therefore, the answer is $4$. The figure below shows all the possible ways, where red stations and railways are administered only by Ko-soku, and blue stations and railways are administered only by Jun-kyu.

Sample Input 2

11 6
???B??A???

Sample Output 2

256

Here, all of the $2^8 = 256$ ways enable us to go from Station $0$ to Station $11$ in no more than $6$ minutes' ride.

Sample Input 3

16 5
?A?B?A?B?A?B?A?

Sample Output 3

10

$10$ ways shown in the figure below enable us to go from Station $0$ to Station $16$ in no more than $5$ minutes' ride.

Sample Input 4

119 15
????????A?????????????????????????BA????????????????????????AB???????A???????????B?????????????????????????????A??????

Sample Output 4

313346281

There are $1623310324709451$ desirable ways. Print this count modulo $(10^9 + 7)$, that is, $313346281$.