A15054. European Trip

The map of Europe can be represented by a set of $n$ cities, numbered from $1$ through $n$ , which are connected by $m$ bidirectional roads, each of which connects two distinct cities. A trip of length $k$ is a sequence of $k+1$ cities $v_1, v_2, \ldots, v_{k+1}$ such that there is a road connecting each consecutive pair $v_i$ , $v_{i+1}$ of cities, for all $1 \le i \le k$ . A special trip is a trip that does not use the same road twice in a row, i.e., a sequence of $k+1$ cities $v_1, v_2, \ldots, v_{k+1}$ such that it forms a trip and $v_i \neq v_{i + 2}$ , for all $1 \le i \le k - 1$ .

Given an integer $k$ , compute the number of distinct special trips of length $k$ which begin and end in the same city. Since the answer might be large, give the answer modulo $998\,244\,353$ .

The first line contains three integers $n$ , $m$ and $k$ ( $3 \le n \le 100$ , $1 \le m \le n(n - 1) / 2$ , $1 \le k \le 10^4$ ) — the number of cities, the number of roads and the length of trips to consider.

Each of the following $m$ lines contains a pair of distinct integers $a$ and $b$ ( $1 \le a, b \le n$ ) — each pair represents a road connecting cities $a$ and $b$ . It is guaranteed that the roads are distinct (i.e., each pair of cities is connected by at most one road).

Print the number of special trips of length $k$ which begin and end in the same city, modulo $998\,244\,353$ .

In the first sample, we are looking for special trips of length $2$ , but since we cannot use the same road twice once we step away from a city we cannot go back, so the answer is $0$ .

In the second sample, we have the following $12$ special trips of length $3$ which begin and end in the same city: $(1, 2, 4, 1)$ , $(1, 3, 4, 1)$ , $(1, 4, 2, 1)$ , $(1, 4, 3, 1)$ , $(2, 1, 4, 2)$ , $(2, 4, 1, 2)$ , $(3, 1, 4, 3)$ , $(3, 4, 1, 3)$ , $(4, 1, 3, 4)$ , $(4, 3, 1, 4)$ , $(4, 1, 2, 4)$ , and $(4, 2, 1, 4)$ .