|LIS| = 3

AtCoder

IDabc237_f

Time2000ms

Memory256MB

Difficulty—

Find the number of sequences that satisfy all of the conditions below, modulo $998244353$. * The length is $N$. * Each of the elements is an integer between $1$ and $M$ (inclusive). * Its longest increasing subsequence has the length of exactly $3$. ## Constraints * $3 \leq N \leq 1000$ * $3 \leq M \leq 10$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ [samples] ## Notes A **subsequence** of a sequence is the result of removing zero or more elements from it and concatenating the remaining elements without changing the order. For example, $(10,30)$ is a subsequence of $(10,20,30)$, while $(20,10)$ is not a subsequence of $(10,20,30)$. A **longest increasing subsequence** of a sequence is its **strictly** increasing subsequence with the greatest length.

Samples

Input #1

4 5

Output #1

135

One sequence that satisfies the conditions is $(3,4,1,5)$.  
On the other hand, $(4,4,1,5)$ do not, because its longest increasing subsequence has the length of $2$.

Input #2

3 4

Output #2

Input #3

111 3

Output #3

144980434

Find the count modulo $998244353$.

API Response (JSON)

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  "problem": {
    "name": "|LIS| = 3",
    "description": {
      "content": "Find the number of sequences that satisfy all of the conditions below, modulo $998244353$. *   The length is $N$. *   Each of the elements is an integer between $1$ and $M$ (inclusive). *   Its longe",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
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    "is_sync": true,
    "sync_url": null,
    "sign": "abc237_f"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Find the number of sequences that satisfy all of the conditions below, modulo $998244353$.\n\n*   The length is $N$.\n*   Each of the elements is an integer between $1$ and $M$ (inclusive).\n*   Its longe...",
      "is_translate": false,
      "language": "English"
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}

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