P15864 T671115 [LBA-OI R3 D] Array Chess

To win the LBA Grand Finals, the coach is designing a brand-new offensive formation.

We are given a basketball court with $n$ horizontal division lines and $m$ vertical division lines, dividing the court into an $n \times m$ grid of regions. The coach assigns a **unique** formation number (from $0$ to $nm-1$) to each region, forming a formation matrix. The **formation diversity value** of a matrix is defined as the number of **distinct values** of the minimum missing formation number across all submatrices of the matrix. ::::info[Example]{open} If the formation numbers in a submatrix are $\{0,1,3,4\}$, the minimum missing number is $2$. If the formation numbers in a submatrix are $\{0,2\}$, the minimum missing number is $1$. :::: Find the sum of the **formation diversity values** over all possible formation matrices. Since the answer can be large, output it modulo $998244353$. --- ### Formal Statement Given $n, m$. An $n \times m$ matrix is valid if it is a permutation of integers from $0$ to $nm-1$. Define $val$ of a matrix as the **size** of the set containing the $\text{mex}$ of all its submatrices. Compute the sum of $val$ over all valid matrices, modulo $998244353$. ::::success[Definition of Submatrix] A submatrix is formed by deleting a prefix of rows/columns and a suffix of rows/columns from the original matrix. ::::

One line with two integers $n, m$.

One integer representing the answer modulo $998244353$.

### Explanation of Sample 1 There are two valid matrices: 1. `0 1` 2. `1 0` For `0 1`, the submatrices are `0` (mex = $1$), `1` (mex = $0$), `0 1` (mex = $2$). The set is $\{0,1,2\}$, so $val = 3$. The case `1 0` is symmetric. Total answer: $3 + 3 = 6$. ### Data Constraints | Subtask | $n \times m$ | Special Property | Score | |:-:|:-:|:-:|:-:| | $1$ | $\le 8$ | A | $10$ | | $2$ | $\le 5 \times 10^6$ | A | $10$ | | $3$ | $\le 225$ | B | $10$ | | $4$ | $\le 6400$ | C | $20$ | | $5$ | $\le 1.6 \times 10^5$ | D | $10$ | | $6$ | $\le 5 \times 10^5$ | None | $20$ | | $7$ | $\le 5 \times 10^6$ | None | $20$ | Special Property A: $n = 1$. Special Property B: $n, m \le 15$. Special Property C: $n, m \le 80$. Special Property D: $n, m \le 400$. For $100\%$ data: $1 \le n \times m \le 5 \times 10^6$. **Please pay attention to the constant complexity of your implementation.**