Similar Permutation

AtCoder

IDabc282_g

Time4000ms

Memory256MB

Difficulty—

Below, a permutation of $(1,2,\ldots,N)$ is simply called a permutation. For two permutations $A=(A_1,A_2,\ldots,A_N),B=(B_1,B_2,\ldots,B_N)$, let us define their **similarity** as the number of integers $i$ between $1$ and $N-1$ such that: * $(A_{i+1}-A_i)(B_{i+1}-B_i)>0$. Find the number, modulo a prime number $P$, of pairs of permutations $(A,B)$ whose similarity is $K$. ## Constraints * $2\leq N \leq 100$ * $0\leq K \leq N-1$ * $10^8 \leq P \leq 10^9$ * $P$ is a prime number. * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $K$ $P$ [samples]

Samples

Input #1

3 1 282282277

Output #1

16

For instance, below is a pair of permutations that satisfies the condition.

*   $A=(1,2,3)$
*   $B=(1,3,2)$

Here, we have $(A_2 - A_1)(B_2 -B_1) > 0$ and $(A_3 - A_2)(B_3 -B_2) < 0$, so the similarity of $A$ and $B$ is $1$.

Input #2

50 25 998244353

Output #2

131276976

Print the number modulo $P$.

API Response (JSON)

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      "content": "Below, a permutation of $(1,2,\\ldots,N)$ is simply called a permutation.\nFor two permutations $A=(A_1,A_2,\\ldots,A_N),B=(B_1,B_2,\\ldots,B_N)$, let us define their **similarity** as the number of integ...",
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