Sort A[i]-i

AtCoder

IDwtf22_day1_e

Time5000ms

Memory256MB

Difficulty—

You are given positive integers $N$ and $M$. Here, $N<M$. A sequence of $N$ non-negative integers $a=(a_1,a_2,\cdots,a_N)$ is said to be a **good** sequence when it satisfies the following condition: * $0 \leq a_1 \leq a_2 \leq \cdots \leq a_N \leq M$. For a good sequence $a$, let $f(a)$ be the sequence resulting from sorting $(a_1-1,a_2-2,\cdots,a_N-N)$ in ascending order. For each $k=1,2,\cdots,N$, solve the following problem. * Assume that we have found the $k$\-th element of $f(a)$ for every possible good sequence $a$. Find the sum, modulo $998244353$, of these values. ※ The answer should be between $0$ and $998244352$ even if the sum is negative. ## Constraints * $1 \leq N < M \leq 10^6$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $M$ [samples]

Samples

Input #1

2 3

Output #1

998244349
4

Here are all possible $a$ and the corresponding $f(a)$.

*   $a=(0,0)$: $f(a)=(-2,-1)$
*   $a=(0,1)$: $f(a)=(-1,-1)$
*   $a=(0,2)$: $f(a)=(-1,0)$
*   $a=(0,3)$: $f(a)=(-1,1)$
*   $a=(1,1)$: $f(a)=(-1,0)$
*   $a=(1,2)$: $f(a)=(0,0)$
*   $a=(1,3)$: $f(a)=(0,1)$
*   $a=(2,2)$: $f(a)=(0,1)$
*   $a=(2,3)$: $f(a)=(1,1)$
*   $a=(3,3)$: $f(a)=(1,2)$

Thus, the sum of the first elements is $-4$, and the sum of the second elements is $4$.

Input #2

3 4

Output #2

998244329
0
24

Input #3

4 6

Output #3

Input #4

10 1000000

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Sort A[i]-i",
    "description": {
      "content": "You are given positive integers $N$ and $M$. Here, $N<M$. A sequence of $N$ non-negative integers $a=(a_1,a_2,\\cdots,a_N)$ is said to be a **good** sequence when it satisfies the following condition: ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "wtf22_day1_e"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "You are given positive integers $N$ and $M$. Here, $N<M$.\nA sequence of $N$ non-negative integers $a=(a_1,a_2,\\cdots,a_N)$ is said to be a **good** sequence when it satisfies the following condition:\n...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments