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G - Similar Permutation

Score : $600$ points

Problem Statement

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:

$N$ $K$ $P$

Output

Print the answer.

3 1 282282277

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 $1$.

50 25 998244353

131276976

Print the number modulo $P$.