Playoff Fixing
Description
$2^k$ teams participate in a playoff tournament. The teams are numbered from $1$ to $2^k$, in order of decreasing strength. So, team $1$ is the strongest one, team $2^k$ is the weakest one. A team with a smaller number always defeats a team with a larger number.
First of all, the teams are arranged in some order during a procedure called seeding. Each team is assigned another unique value from $1$ to $2^k$, called a seed, that represents its starting position in the playoff.
The tournament consists of $2^k - 1$ games. They are held as follows: the teams are split into pairs: team with seed $1$ plays against team with seed $2$, team with seed $3$ plays against team with seed $4$ (exactly in this order), and so on (so, $2^{k-1}$ games are played in that phase). When a team loses a game, it is eliminated.
After that, only $2^{k-1}$ teams remain. If only one team remains, it is declared the champion; otherwise, $2^{k-2}$ games are played: in the first one of them, the winner of the game "seed $1$ vs seed $2$" plays against the winner of the game "seed $3$ vs seed $4$", then the winner of the game "seed $5$ vs seed $6$" plays against the winner of the game "seed $7$ vs seed $8$", and so on. This process repeats until only one team remains.
After the tournament ends, the teams are assigned places according to the tournament phase when they were eliminated. In particular:
- the winner of the tournament gets place $1$;
- the team eliminated in the finals gets place $2$;
- both teams eliminated in the semifinals get place $3$;
- all teams eliminated in the quarterfinals get place $5$;
- all teams eliminated in the 1/8 finals get place $9$, and so on.
Now that we established the rules, we do a little rigging. In particular, we want:
- team $1$ (not team with seed $1$) to take place $1$;
- team $2$ to take place $2$;
- teams $3$ and $4$ to take place $3$;
- teams from $5$ to $8$ to take place $5$, and so on.
For example, this picture describes one of the possible ways the tournament can go with $k = 3$, and the resulting places of the teams:
Some seeds are already reserved for some teams (we are not the only ones rigging the tournament, apparently). We have to fill the rest of the seeds with the remaining teams to achieve the desired placements. How many ways are there to do that? Since that value might be large, print it modulo $998\,244\,353$.
题意简述
有 支队伍打淘汰赛。我们按照队伍的强弱编号，编号为 的队伍是最强的，而编号为 的队伍是最弱的。
你将这 支队伍从左至右排成一排。
淘汰赛会进行 轮。每轮淘汰赛，这一排中的第一支队伍和第二支队伍对决，第三只队伍和第四只队伍对决，等。这些对决同时进行，实力较弱的队伍将在对决中被淘汰，从这一排队伍中删除。一轮淘汰赛后，这一排队伍的长度会减半。
有些队伍内定了在这一排队伍中的位置，而其它队伍在排中的位置由你来决定。请问你有多少种安排位置的方案，满足在淘汰赛的第 轮中，被淘汰的队伍一定是当前这一排中最弱的 支队伍？
请输出答案对 取模后的值。
输入
输入第一行包括一个整数 表示有 支队伍；
输入第二行包括 个整数 。如果 ，说明排中第 个位置没有被内定；否则，说明排中第 个位置被队伍 内定了。
输出
输出方案数对 取模后的值。
Input
The first line contains a single integer $k$ ($0 \le k \le 19$) — there are $2^k$ teams.
The second line contains $2^k$ integers $a_1, a_2, \dots, a_{2^k}$ ($a_i = -1$ or $1 \le a_i \le 2^k$). If $a_i \ne -1$, then team $a_i$ has seed $i$. Otherwise, the seed $i$ is not reserved for any team.
All values, that are not $-1$, are distinct.
Output
Print a single integer — the number of ways to fill the non-reserved seeds so that the tournament goes as planned, modulo $998\,244\,353$.
2
1 2 3 4
2
1 3 4 2
1
-1 -1
2
-1 -1 -1 -1
3
-1 -1 -1 -1 2 -1 -1 -1
0
1
0
1
2
16
768
1
相关
在下列比赛中: