Score: $650$ points

问题描述

给定16个非负整数 $X_{i, j, k, l}$ $(i, j, k, l \in \lbrace 0, 1 \rbrace)$，按照 $(i, j, k, l)$ 的升序排列。

令 $N = \displaystyle \sum_{i=0}^1 \sum_{j=0}^1 \sum_{k=0}^1 \sum_{l=0}^1 X_{i,j,k,l}$。

求满足以下条件的长度为 $N+3$、仅包含0和1的序列 $(A_1, A_2, ..., A_{N+3})$ 的数量，模 $998244353$ 后的结果。

• 对于所有四元组 $(i, j, k, l)$ $(i, j, k, l \in \lbrace 0, 1 \rbrace)$，存在恰好 $X_{i,j,k,l}$ 个整数 $s$，满足 $1 \leq s \leq N$，且：
  • $A_s = i$，$A_{s + 1} = j$，$A_{s + 2} = k$，以及 $A_{s + 3} = l$。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given $16$ non-negative integers $X_{i, j, k, l}$ $(i, j, k, l \in \lbrace 0, 1 \rbrace)$ in ascending order of $(i, j, k, l)$.
Let $N = \displaystyle \sum_{i=0}^1 \sum_{j=0}^1 \sum_{k=0}^1 \sum_{l=0}^1 X_{i,j,k,l}$.
Find the number, modulo $998244353$, of sequences $(A_1, A_2, ..., A_{N+3})$ of length $N + 3$ consisting of $0$s and $1$s that satisfy the following condition.

• For every quadruple of integers $(i, j, k, l)$ $(i, j, k, l \in \lbrace 0, 1 \rbrace)$, there are exactly $X_{i,j,k,l}$ integers $s$ between $1$ and $N$, inclusive, that satisfy:
  • $A_s = i$, $A_{s + 1} = j$, $A_{s + 2} = k$, and $A_{s + 3} = l$.

Constraints

• $X_{i, j, k, l}$ are all non-negative integers.
• $1 \leq \displaystyle \sum_{i=0}^1 \sum_{j=0}^1 \sum_{k=0}^1 \sum_{l=0}^1 X_{i,j,k,l} \leq 10^6$

Input

The input is given from Standard Input in the following format:

$X_{0,0,0,0}$ $X_{0,0,0,1}$ $X_{0,0,1,0}$ $X_{0,0,1,1}$ $X_{0,1,0,0}$ $X_{0,1,0,1}$ $X_{0,1,1,0}$ $X_{0,1,1,1}$ $X_{1,0,0,0}$ $X_{1,0,0,1}$ $X_{1,0,1,0}$ $X_{1,0,1,1}$ $X_{1,1,0,0}$ $X_{1,1,0,1}$ $X_{1,1,1,0}$ $X_{1,1,1,1}$

Output

Print the number, modulo $998244353$, of sequences that satisfy the condition in the problem statement.

Sample Input 1

0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0

Sample Output 1

1

This input corresponds to the case where $X_{1, 0, 1, 0}$ and $X_{1, 1, 0, 1}$ are $1$ and all others are $0$.
In this case, only one sequence satisfies the condition, which is $(1, 1, 0, 1, 0)$.

Sample Input 2

1 1 2 0 1 2 1 1 1 1 1 2 1 0 1 0

Sample Output 2

16

Sample Input 3

21 3 3 0 3 0 0 0 4 0 0 0 0 0 0 0

Sample Output 3

2024

Sample Input 4

62 67 59 58 58 69 57 66 67 50 68 65 59 64 67 61

Sample Output 4

741536606

update @ 2024/3/10 01:26:33