A13760. Boboniu Walks on Graph

Boboniu has a directed graph with $n$ vertices and $m$ edges.

The out-degree of each vertex is at most $k$ .

Each edge has an integer weight between $1$ and $m$ . No two edges have equal weights.

Boboniu likes to walk on the graph with some specific rules, which is represented by a tuple $(c_1,c_2,\ldots,c_k)$ . If he now stands on a vertex $u$ with out-degree $i$ , then he will go to the next vertex by the edge with the $c_i$ -th $(1\le c_i\le i)$ smallest weight among all edges outgoing from $u$ .

Now Boboniu asks you to calculate the number of tuples $(c_1,c_2,\ldots,c_k)$ such that

- $1\le c_i\le i$ for all $i$ ( $1\le i\le k$ ).
- Starting from any vertex $u$ , it is possible to go back to $u$ in finite time by walking on the graph under the described rules.

The first line contains three integers $n$ , $m$ and $k$ ( $2\le n\le 2\cdot 10^5$ , $2\le m\le \min(2\cdot 10^5,n(n-1) )$ , $1\le k\le 9$ ).

Each of the next $m$ lines contains three integers $u$ , $v$ and $w$ $(1\le u,v\le n,u\ne v,1\le w\le m)$ , denoting an edge from $u$ to $v$ with weight $w$ . It is guaranteed that there are no self-loops or multiple edges and each vertex has at least one edge starting from itself.

It is guaranteed that the out-degree of each vertex is at most $k$ and no two edges have equal weight.

Print one integer: the number of tuples.

For the first example, there are two tuples: $(1,1,3)$ and $(1,2,3)$ . The blue edges in the picture denote the $c_i$ -th smallest edges for each vertex, which Boboniu chooses to go through.

For the third example, there's only one tuple: $(1,2,2,2)$ .

The out-degree of vertex $u$ means the number of edges outgoing from $u$ .