A < AP

AtCoder

IDarc151_b

Time2000ms

Memory256MB

Difficulty—

You are given a permutation $P = (P_1, P_2, \ldots, P_N)$ of $(1, 2, \ldots, N)$. Print the number of integer sequences $A = (A_1, A_2, \ldots, A_N)$ of length $N$ that satisfy both of the conditions below, modulo $998244353$. * $1 \leq A_i \leq M$ for every $i = 1, 2, \ldots, N$. * The integer sequence $A$ is lexicographically smaller than the integer sequence $(A_{P_1}, A_{P_2}, \ldots, A_{P_N})$. What is lexicographical order?An integer sequence $X = (X_1,X_2,\ldots,X_N)$ is **lexicographically smaller** than an integer sequence $Y = (Y_1,Y_2,\ldots,Y_N)$ when there is an integer $1 \leq i \leq N$ that satisfies both of the conditions below. * For all integers $j$ ($1 \leq j \leq i-1$), $X_j=Y_j$. * $X_i < Y_i$ ## Constraints * $1 \leq N \leq 2 \times 10^5$ * $1 \leq M \leq 10^9$ * $1 \leq P_i \leq N$ * $i \neq j \implies P_i \neq P_j$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ $P_1$ $P_2$ $\ldots$ $P_N$ [samples]

Samples

Input #1

4 2
4 1 3 2

Output #1

6

Six integer sequences $A$ satisfy both of the conditions: $(1, 1, 1, 2)$, $(1, 1, 2, 2)$, $(1, 2, 1, 2)$, $(1, 2, 2, 2)$, $(2, 1, 1, 2)$, and $(2, 1, 2, 2)$.  
For instance, when $A = (1, 1, 1, 2)$, we have $(A_{P_1}, A_{P_2}, A_{P_3}, A_{P_4}) = (2, 1, 1, 1)$, which is lexicographically larger than $A$.

Input #2

1 1
1

Output #2

Input #3

20 100000
11 15 3 20 17 6 1 9 5 19 10 16 7 8 12 2 18 14 4 13

Output #3

55365742

API Response (JSON)

{
  "problem": {
    "name": "A < AP",
    "description": {
      "content": "You are given a permutation $P = (P_1, P_2, \\ldots, P_N)$ of $(1, 2, \\ldots, N)$. Print the number of integer sequences $A = (A_1, A_2, \\ldots, A_N)$ of length $N$ that satisfy both of the conditions ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc151_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given a permutation $P = (P_1, P_2, \\ldots, P_N)$ of $(1, 2, \\ldots, N)$.\nPrint the number of integer sequences $A = (A_1, A_2, \\ldots, A_N)$ of length $N$ that satisfy both of the conditions ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments