CF1909I.Short Permutation Problem

Xomu - Last Dance

⠀

You are given an integer $n$ .

For each $(m, k)$ such that $3 \leq m \leq n+1$ and $0 \leq k \leq n-1$ , count the permutations of $[1, 2, ..., n]$ such that $p_i + p_{i+1} \geq m$ for exactly $k$ indices $i$ , modulo $998\,244\,353$ .

The input consists of a single line, which contains two integers $n$ , $x$ ( $2 \leq n \leq 4000$ , $1 \leq x < 1\,000\,000\,007$ ).

Let $a_{m,k}$ be the answer for the pair $(m, k)$ , modulo $998\,244\,353$ .

Let $$$$\large S = \sum_{m=3}^{n+1} \sum_{k=0}^{n-1} a_{m,k}x^{mn+k}\phantom{0}. $$

Output a single line with an integer: $S$ modulo $1\\,000\\,000\\,007$ .

Note that using two different modulos is intentional. We want you to calculate all the $a\_{m,k}$ modulo $998\\,244\\,353$ , then treat them like integers in the range \[0, 998\\,244\\,352\] , and hash them modulo $1\\,000\\,000\\,007$$$.

In the first test case, the answers for all $(m, k)$ are shown in the following table:

$k = 0$ $k = 1$ $k = 2$ $m = 3$ $0$ $0$ $6$ $m = 4$ $0$ $4$ $2$ - The answer for $(m, k) = (3, 2)$ is $6$ , because for every permutation of length $3$ , $a_i + a_{i+1} \geq 3$ exactly $2$ times.

• The answer for $(m, k) = (4, 2)$ is $2$ . In fact, there are $2$ permutations of length $3$ such that $a_i + a_{i+1} \geq 4$ exactly $2$ times: $[1, 3, 2]$ , $[2, 3, 1]$ .

Therefore, the value to print is $2^9 \cdot 0 + 2^{10} \cdot 0 + 2^{11} \cdot 6 + 2^{12} \cdot 0 + 2^{13} \cdot 4 + 2^{14} \cdot 2 \equiv 77\,824 \phantom{0} (\text{mod} \phantom{0} 1\,000\,000\,007)$ .

In the second test case, the answers for all $(m, k)$ are shown in the following table:

$k = 0$ $k = 1$ $k = 2$ $k = 3$ $m = 3$ $0$ $0$ $0$ $24$ $m = 4$ $0$ $0$ $12$ $12$ $m = 5$ $0$ $4$ $16$ $4$ - The answer for $(m, k) = (5, 1)$ is $4$ . In fact, there are $4$ permutations of length $4$ such that $a_i + a_{i+1} \geq 5$ exactly $1$ time: $[2, 1, 3, 4]$ , $[3, 1, 2, 4]$ , $[4, 2, 1, 3]$ , $[4, 3, 1, 2]$ .

In the third test case, the answers for all $(m, k)$ are shown in the following table:

$k = 0$ $k = 1$ $k = 2$ $k = 3$ $k = 4$ $k = 5$ $k = 6$ $k = 7$ $m = 3$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $40320$ $m = 4$ $0$ $0$ $0$ $0$ $0$ $0$ $10080$ $30240$ $m = 5$ $0$ $0$ $0$ $0$ $0$ $1440$ $17280$ $21600$ $m = 6$ $0$ $0$ $0$ $0$ $480$ $8640$ $21600$ $9600$ $m = 7$ $0$ $0$ $0$ $96$ $3456$ $16416$ $16896$ $3456$ $m = 8$ $0$ $0$ $48$ $2160$ $12960$ $18240$ $6480$ $432$ $m = 9$ $0$ $16$ $1152$ $9648$ $18688$ $9648$ $1152$ $16$