Reverse and Inversion

AtCoder

IDarc154_e

Time2000ms

Memory256MB

Difficulty—

For a permutation $Q=(Q_1,Q_2,\dots,Q_N)$ of $(1,2,\dots,N)$, let $f(Q)$ be the following value: > the sum of $j-i$ over all pairs of integers $(i,j)$ such that $1 \le i < j \le N$ and $Q_i > Q_j$. You are given a permutation $P=(P_1,P_2,\dots,P_N)$ of $(1,2,\dots,N)$. Let us repeat the following operation $M$ times. * Choose a pair of integers $(i,j)$ such that $1 \le i \le j \le N$. Reverse $P_i,P_{i+1},\dots,P_j$. Formally, replace the values of $P_i,P_{i+1},\dots,P_j$ with $P_j,P_{j-1},\dots,P_i$ simultaneously. There are $\left(\frac{N(N+1)}{2}\right)^{M}$ ways to repeat the operation. Assume that we have computed $f(P)$ for all those ways. Find the sum of these $\left(\frac{N(N+1)}{2}\right)^{M}$ values, modulo $998244353$. ## Constraints * $1 \le N,M \le 2 \times 10^5$ * $(P_1,P_2,\dots,P_N)$ is a permutation of $(1,2,\dots,N)$. ## Input The input is given from Standard Input in the following format: $N$ $M$ $P_1$ $P_2$ $\dots$ $P_N$ [samples]

Samples

Input #1

2 1
1 2

Output #1

1

There are three ways to perform the operation, as follows.

*   Choose $(i,j)=(1,1)$, making $P=(1,2)$, where $f(P)=0$.
*   Choose $(i,j)=(1,2)$, making $P=(2,1)$, where $f(P)=1$.
*   Choose $(i,j)=(2,2)$, making $P=(1,2)$, where $f(P)=0$.

Thus, the answer is $0+1+0=1$.

Input #2

3 2
3 2 1

Output #2

Input #3

10 2023
5 8 1 9 3 10 4 7 2 6

Output #3

543960046

API Response (JSON)

{
  "problem": {
    "name": "Reverse and Inversion",
    "description": {
      "content": "For a permutation $Q=(Q_1,Q_2,\\dots,Q_N)$ of $(1,2,\\dots,N)$, let $f(Q)$ be the following value: > the sum of $j-i$ over all pairs of integers $(i,j)$ such that $1 \\le i < j \\le N$ and $Q_i > Q_j$. ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc154_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "For a permutation $Q=(Q_1,Q_2,\\dots,Q_N)$ of $(1,2,\\dots,N)$, let $f(Q)$ be the following value:\n\n> the sum of $j-i$ over all pairs of integers $(i,j)$ such that $1 \\le i < j \\le N$ and $Q_i > Q_j$.\n\n...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments