Ex - Yet Another Path Counting

AtCoder

IDabc259_h

Time2000ms

Memory256MB

Difficulty—

We have a grid of squares with $N$ rows arranged vertically and $N$ columns arranged horizontally. The square at the $i$\-th row from the top and $j$\-th column from the left has an integer label $a_{i,j}$. Consider paths obtained by starting on one of the squares and going **right or down** to an adjacent square zero or more times. Find the number, modulo $998244353$, of such paths that start and end on squares with the same label. Two paths are distinguished when they visit different sets of squares (including the starting and ending squares). ## Constraints * $1 \leq N \leq 400$ * $1 \leq a_{i,j} \leq N^2$ * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $a_{1,1}$ $\ldots$ $a_{1,N}$ $\vdots$ $a_{N,1}$ $\ldots$ $a_{N,N}$ [samples]

Samples

Input #1

2
1 3
3 1

Output #1

6

The following six paths satisfy the requirements. ($(i, j)$ denotes the square at the $i$\-th row from the top and $j$\-th column from the left. Each path is represented as a sequence of squares it visits.)

*   $(1,1)$
*   $(1,1)$ → $(1,2)$ → $(2,2)$
*   $(1,1)$ → $(2,1)$ → $(2,2)$
*   $(1,2)$
*   $(2,1)$
*   $(2,2)$

API Response (JSON)

{
  "problem": {
    "name": "Ex - Yet Another Path Counting",
    "description": {
      "content": "We have a grid of squares with $N$ rows arranged vertically and $N$ columns arranged horizontally. The square at the $i$\\-th row from the top and $j$\\-th column from the left has an integer label $a_{",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc259_h"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a grid of squares with $N$ rows arranged vertically and $N$ columns arranged horizontally. The square at the $i$\\-th row from the top and $j$\\-th column from the left has an integer label $a_{...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments