Split and Maximize

AtCoder

IDarc145_c

Time2000ms

Memory256MB

Difficulty—

The score of a permutation $P=(P_1,P_2,\ldots,P_{2N})$ of $(1,2,\ldots,2N)$ is defined as follows: > Consider dividing $P$ into two (not necessarily contiguous) subsequences $A = (A_1,A_2,\ldots,A_N)$ and $B = (B_1,B_2,\ldots,B_N)$. The score of $P$ is the maximum possible value of $\displaystyle\sum_{i=1}^{N}A_i B_i$ in such a division. Let $M$ be the maximum among the scores of all permutations of $(1,2,\ldots,2N)$. Find the number, modulo $998244353$, of permutations of $(1,2,\ldots,2N)$ with the score of $M$. ## Constraints * $1 \leq N \leq 2\times 10^5$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ [samples]

Samples

Input #1

Output #1

16

The maximum among the scores of the $24$ possible permutations, $M$, is $14$, and there are $16$ permutations with the score of $14$.
For instance, the permutation $(1,2,3,4)$ achieves $\sum _{i=1}^{N}A_i B_i = 14$ in the division $A=(1,3), B=(2,4)$.

Input #2

Output #2

391163238

Print the count modulo $998244353$.

API Response (JSON)

{
  "problem": {
    "name": "Split and Maximize",
    "description": {
      "content": "The score of a permutation $P=(P_1,P_2,\\ldots,P_{2N})$ of $(1,2,\\ldots,2N)$ is defined as follows: > Consider dividing $P$ into two (not necessarily contiguous) subsequences $A = (A_1,A_2,\\ldots,A_N)",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc145_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "The score of a permutation $P=(P_1,P_2,\\ldots,P_{2N})$ of $(1,2,\\ldots,2N)$ is defined as follows:\n\n> Consider dividing $P$ into two (not necessarily contiguous) subsequences $A = (A_1,A_2,\\ldots,A_N)...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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