codeforces#P1874E. Jellyfish and Hack
Jellyfish and Hack
Description
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:
function fun(A)
if A.length > 0
let L[1 ... L.length] and R[1 ... R.length] be new arrays
L.length = R.length = 0
for i = 2 to A.length
if A[i] < A[1]
L.length = L.length + 1
L[L.length] = A[i]
else
R.length = R.length + 1
R[R.length] = A[i]
return A.length + fun(L) + fun(R)
else
return 0
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 $10^9+7$.
Input
The only line of the input contains two integers $n$ and $lim$ ($1 \leq n \leq 200$, $1 \leq lim \leq 10^9$).
Output
Output the number of different permutations that satisfy the condition modulo $10^9+7$.
4 10
8 32
8
1280
Note
In the first example, $P = [1, 4, 2, 3]$ satisfies the condition, because: $\mathrm{fun([1, 4, 2, 3]) = 4 + fun([4, 2, 3]) = 7 + fun([2, 3]) = 9 + fun([3]) = 10}$
Do remember to output the answer modulo $10^9+7$.