201322: [AtCoder]ARC132 C - Almost Sorted

Problem Statement

Given is a sequence $a_1,\dots,a_n$ consisting of $1,\dots, n$ and $-1$, along with an integer $d$. How many sequences $p_1,\dots,p_n$ satisfy the conditions below? Print the count modulo $998244353$.

• $p_1,\dots,p_n$ is a permutation of $1,\dots, n$.
• For each $i=1,\dots,n$, $a_i=p_i$ if $a_i\neq -1$. (That is, $p_1,\dots,p_n$ can be obtained by replacing the $-1$s in $a_1,\dots,a_n$ in some way.)
• For each $i=1,\dots,n$, $|p_i - i|\le d$.

Constraints

• $1 \le d \le 5$
• $d < n \le 500$
• $1\le a_i \le n$ or $a_i=-1$.
• $|a_i-i|\le d$ if $a_i\neq -1$.
• $a_i\neq a_j$, if $i\neq j$ and $a_i, a_j \neq -1$.
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$n$ $d$
$a_1$ $\dots$ $a_n$

Output

Print the number of sequences that satisfy the conditions, modulo $998244353$.

Sample Input 1

4 2
3 -1 1 -1

Sample Output 1

2

The conditions are satisfied by $(3,2,1,4)$ and $(3,4,1,2)$.

Sample Input 2

5 1
2 3 4 5 -1

Sample Output 2

0

The only permutation of $1,2,3,4,5$ that can be obtained by replacing the $-1$ is $(2,3,4,5,1)$, whose fifth term violates the condition, so the answer is $0$.

Sample Input 3

16 5
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

Sample Output 3

794673086

Print the count modulo $998244353$.