Sum of CC

AtCoder

IDarc187_b

Time2000ms

Memory256MB

Difficulty—

For a sequence $A = (A_1, \ldots, A_N)$ of length $N$, define $f(A)$ as follows. * Prepare a graph with $N$ vertices labeled $1$ to $N$ and zero edges. For every integer pair $(i, j)$ satisfying $1 \leq i < j \leq N$, if $A_i \leq A_j$, draw a bidirectional edge connecting vertices $i$ and $j$. Define $f(A)$ as the number of connected components in the resulting graph. You are given a sequence $B = (B_1, \ldots, B_N)$ of length $N$. Each element of $B$ is $-1$ or an integer between $1$ and $M$, inclusive. By replacing every occurrence of $-1$ in $B$ with an integer between $1$ and $M$, one can obtain $M^q$ sequences $B'$, where $q$ is the number of $-1$ in $B$. Find the sum, modulo $998244353$, of $f(B')$ over all possible $B'$. ## Constraints * All input numbers are integers. * $2 \leq N \leq 2000$ * $1 \leq M \leq 2000$ * Each $B_i$ is $-1$ or an integer between $1$ and $M$, inclusive. ## Input The input is given from Standard Input in the following format: $N$ $M$ $B_1$ $\ldots$ $B_N$ [samples]

Samples

Input #1

3 3
2 -1 1

Output #1

6

There are three possible sequences $B'$: $(2,1,1)$, $(2,2,1)$, and $(2,3,1)$.
When $B' = (2,1,1)$, an edge is drawn only between vertices $2$ and $3$, so the number of connected components is $2$. Thus, $f(B') = 2$.
Similarly, $f(B') = 2$ for $B' = (2,2,1)$ and $f(B') = 2$ for $B' = (2,3,1)$, so the answer is $2 + 2 + 2 = 6$.

Input #2

10 8
-1 7 -1 -1 -1 2 -1 1 -1 2

Output #2

Input #3

11 12
-1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1

Output #3

529513150

Remember to find the sum modulo $998244353$.

API Response (JSON)

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      "content": "For a sequence $A = (A_1, \\ldots, A_N)$ of length $N$, define $f(A)$ as follows.\n\n*   Prepare a graph with $N$ vertices labeled $1$ to $N$ and zero edges. For every integer pair $(i, j)$ satisfying $1...",
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