A16216. Jellyfish and Hack

It is well known that quick sort works by randomly selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. But Jellyfish thinks that choosing a random element is just a waste of time, so she always chooses the first element to be the pivot. The time her code needs to run can be calculated by the following pseudocode:

```
<pre class="verbatim"><br></br>function fun(A)<br></br> if A.length > 0<br></br> let L[1 ... L.length] and R[1 ... R.length] be new arrays<br></br> L.length = R.length = 0<br></br> for i = 2 to A.length<br></br> if A[i] < A[1]<br></br> L.length = L.length + 1<br></br> L[L.length] = A[i]<br></br> else<br></br> R.length = R.length + 1<br></br> R[R.length] = A[i]<br></br> return A.length + fun(L) + fun(R)<br></br> else<br></br> return 0<br></br>
```

Now you want to show her that her code is slow. When the function $\mathrm{fun(A)}$ is greater than or equal to $lim$ , her code will get $\text{Time Limit Exceeded}$ . You want to know how many distinct permutations $P$ of $[1, 2, \dots, n]$ satisfies $\mathrm{fun(P)} \geq lim$ . Because the answer may be large, you will only need to find the answer modulo $10^9+7$ .

The only line of the input contains two integers $n$ and $lim$ ( $1 \leq n \leq 200$ , $1 \leq lim \leq 10^9$ ).

Output the number of different permutations that satisfy the condition modulo $10^9+7$ .

In the first example, $P = [1, 4, 2, 3]$ satisfies the condition, because: $\mathrm{fun(4, [1, 4, 2, 3]) = 4 + fun(3, [4, 2, 3]) = 7 + fun(2, [2, 3]) = 9 + fun(1, [3]) = 10}$

Do remember to output the answer modulo $10^9+7$ .