Trinity

AtCoder

IDagc021_f

Time6000ms

Memory256MB

Difficulty—

We have an $N \times M$ grid. The square at the $i$\-th row and $j$\-th column will be denoted as $(i,j)$. Particularly, the top-left square will be denoted as $(1,1)$, and the bottom-right square will be denoted as $(N,M)$. Takahashi painted some of the squares (possibly zero) black, and painted the other squares white. We will define an integer sequence $A$ of length $N$, and two integer sequences $B$ and $C$ of length $M$ each, as follows: * $A_i(1\leq i\leq N)$ is the minimum $j$ such that $(i,j)$ is painted black, or $M+1$ if it does not exist. * $B_i(1\leq i\leq M)$ is the minimum $k$ such that $(k,i)$ is painted black, or $N+1$ if it does not exist. * $C_i(1\leq i\leq M)$ is the maximum $k$ such that $(k,i)$ is painted black, or $0$ if it does not exist. How many triples $(A,B,C)$ can occur? Find the count modulo $998244353$. ## Constraints * $1 \leq N \leq 8000$ * $1 \leq M \leq 200$ * $N$ and $M$ are integers. ## Input Input is given from Standard Input in the following format: $N$ $M$ ## Output Print the number of triples $(A,B,C)$, modulo $998244353$. [samples] ## Notice In this problem, the length of your source code must be at most $20000$ B. Note that we will invalidate submissions that exceed the maximum length.

Samples

Input #1

2 3

Output #1

64

Since $N=2$, given $B_i$ and $C_i$, we can uniquely determine the arrangement of black squares in each column. For each $i$, there are four possible pairs $(B_i,C_i)$: $(1,1)$, $(1,2)$, $(2,2)$ and $(3,0)$. Thus, the answer is $4^M=64$.

Input #2

4 3

Output #2

Input #3

17 13

Output #3

229876268

Input #4

5000 100

Output #4

57613837

API Response (JSON)

{
  "problem": {
    "name": "Trinity",
    "description": {
      "content": "We have an $N \\times M$ grid. The square at the $i$\\-th row and $j$\\-th column will be denoted as $(i,j)$. Particularly, the top-left square will be denoted as $(1,1)$, and the bottom-right square wil",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 6000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc021_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have an $N \\times M$ grid. The square at the $i$\\-th row and $j$\\-th column will be denoted as $(i,j)$. Particularly, the top-left square will be denoted as $(1,1)$, and the bottom-right square wil...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments