F1. Neko Rules the Catniverse (Small Version)
bitmasks; dp; matrices
This problem is same as the next one, but has smaller constraints.Aki is playing a new video game. In the video game, he will control Neko, the giant cat, to fly between planets in the Catniverse.There are \(n\) planets in the Catniverse, numbered from \(1\) to \(n\). At the beginning of the game, Aki chooses the planet where Neko is initially located. Then Aki performs \(k - 1\) moves, where in each move Neko is moved from the current planet \(x\) to some other planet \(y\) such that: Planet \(y\) is not visited yet. \(1 \leq y \leq x + m\) (where \(m\) is a fixed constant given in the input) This way, Neko will visit exactly \(k\) different planets. Two ways of visiting planets are called different if there is some index \(i\) such that the \(i\)-th planet visited in the first way is different from the \(i\)-th planet visited in the second way.What is the total number of ways to visit \(k\) planets this way? Since the answer can be quite large, print it modulo \(10^9 + 7\).
The only line contains three integers \(n\), \(k\) and \(m\) (\(1 \le n \le 10^5\), \(1 \le k \le \min(n, 12)\), \(1 \le m \le 4\)) — the number of planets in the Catniverse, the number of planets Neko needs to visit and the said constant \(m\).
Print exactly one integer — the number of different ways Neko can visit exactly \(k\) planets. Since the answer can be quite large, print it modulo \(10^9 + 7\).
In the first example, there are \(4\) ways Neko can visit all the planets: \(1 \rightarrow 2 \rightarrow 3\) \(2 \rightarrow 3 \rightarrow 1\) \(3 \rightarrow 1 \rightarrow 2\) \(3 \rightarrow 2 \rightarrow 1\) In the second example, there are \(9\) ways Neko can visit exactly \(2\) planets: \(1 \rightarrow 2\) \(2 \rightarrow 1\) \(2 \rightarrow 3\) \(3 \rightarrow 1\) \(3 \rightarrow 2\) \(3 \rightarrow 4\) \(4 \rightarrow 1\) \(4 \rightarrow 2\) \(4 \rightarrow 3\) In the third example, with \(m = 4\), Neko can visit all the planets in any order, so there are \(5! = 120\) ways Neko can visit all the planets.In the fourth example, Neko only visit exactly \(1\) planet (which is also the planet he initially located), and there are \(100\) ways to choose the starting planet for Neko.