Sequence Scores

AtCoder

IDarc114_c

Time2000ms

Memory256MB

Difficulty—

For a sequence $A$ of length $N$ consisting of integers between $1$ and $M$ (inclusive), let us define $f(A)$ as follows: * We have a sequence $X$ of length $N$, where every element is initially $0$. $f(A)$ is the minimum number of operations required to make $X$ equal $A$ by repeating the following operation: * Specify $1 \leq l \leq r \leq N$ and $1 \leq v \leq M$, then replace $X_i$ with $\max(X_i, v)$ for each $l \leq i \leq r$. Find the sum, modulo $998244353$, of $f(A)$ over all $M^N$ sequences that can be $A$. ## Constraints * $1 \leq N, M \leq 5000$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

2 3

Output #1

15

The $3 ^ 2 = 9$ sequences and the value of $f$ for those are as follows:

*   For $A = (1, 1)$, we can make $X$ equal it with one operation with $(l = 1, r = 2, v = 1)$, so $f(A) = 1$.
*   For $A = (1, 2)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 1)$ and $(l = 2, r = 2, v = 2)$, so $f(A) = 2$.
*   For $A = (1, 3)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 1)$ and $(l = 2, r = 2, v = 3)$, so $f(A) = 2$.
*   For $A = (2, 1)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 1)$ and $(l = 1, r = 1, v = 2)$, so $f(A) = 2$.
*   For $A = (2, 2)$, we can make $X$ equal it with one operation with $(l = 1, r = 2, v = 2)$, so $f(A) = 1$.
*   For $A = (2, 3)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 2)$ , $(l = 2, r = 2, v = 3)$, so $f(A) = 2$.
*   For $A = (3, 1)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 1)$ and $(l = 1, r = 1, v = 3)$ so $f(A) = 2$.
*   For $A = (3, 2)$, we can make $X$ equal it with two operations with $(l = 1, r = 2, v = 2)$ and $(l = 1, r = 1, v = 3)$, so $f(A) = 2$.
*   For $A = (3, 3)$, we can make $X$ equal it with one operation with $(l = 1, r = 2, v = 3)$, so $f(A) = 1$.

The sum of these values is $3 \times 1 + 6 \times 2 = 15$.

Input #2

3 2

Output #2

Input #3

34 56

Output #3

649717324

API Response (JSON)

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    "name": "Sequence Scores",
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      "content": "For a sequence $A$ of length $N$ consisting of integers between $1$ and $M$ (inclusive), let us define $f(A)$ as follows: *   We have a sequence $X$ of length $N$, where every element is initially $0",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
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    "sync_url": null,
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For a sequence $A$ of length $N$ consisting of integers between $1$ and $M$ (inclusive), let us define $f(A)$ as follows:\n\n*   We have a sequence $X$ of length $N$, where every element is initially $0...",
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