Cartesian Tree

A sequence $A$ of distinct integers determines exactly one Cartesian tree. A Cartesian tree is a tree that meets these four conditions.

1. It is a rooted binary tree.
2. Each node corresponds to one element of $A$.
3. An in-order traversal of the tree visits the values in the same order as $A$.
4. The value of a node is smaller than the values of its children, so the tree is a min-heap.

The picture below is the Cartesian tree built from $A = [9, 3, 7, 1, 8, 12, 10, 20, 15, 18, 5]$.

Let $T$ be the Cartesian tree built from $A$. The score of $T$ is computed as follows. For every node with two children, find the positions in $A$ of the two child values and take the difference of those positions. The score of $T$ is the sum of that difference over all such nodes.

In the picture the nodes with two children are 1, 3, 10, 15. The two children of node 1 are 3 and 5, which sit at positions 2 and 11 of $A$, so this node scores $11 - 2 = 9$. The other three nodes score 2, 3 and 2, so the score of $T$ is $9 + 2 + 3 + 2 = 16$.

You are given $N$, $S$ and $\text{MOD}$. There are $N!$ permutations of the numbers $1$ through $N$, and each permutation builds one Cartesian tree. Let $X$ be the number of those trees whose score is at most $S$. Write a program that prints $X$ modulo $\text{MOD}$.

The first line contains $N$, $S$ and $\text{MOD}$, separated by spaces. ($1 \le N \le 100$, $0 \le S \le 100$, $3 \le \text{MOD} \le 10^9$, $\text{MOD}$ is prime)

Print on the first line how many of the $N!$ permutations build a tree whose score is at most $S$, modulo $\text{MOD}$.

For $N = 3$, the permutations $(2, 1, 3)$ and $(3, 1, 2)$ score 2, and the other four permutations score 0.