629. K Inverse Pairs Array

Given two integers 

n

 and 

k

,
find how many different arrays consist of numbers from 

1

 to 

n

 such
that there are exactly 

k

 inverse pairs.

We define an inverse pair as following: For 

ith

 and 

jth

 element
in the array, if 

i

 < 

j

 and 

a[i]

 > 

a[j]

 then
it's an inverse pair; Otherwise, it's not.

Since the answer may be very large, the answer should be modulo 109 + 7.

Example 1:

Input: n = 3, k = 0
Output: 1
Explanation:
Only the array [1,2,3] which consists of numbers from 1 to 3 has exactly 0 inverse pair.

Example 2:

Input: n = 3, k = 1
Output: 2
Explanation:
The array [1,3,2] and [2,1,3] have exactly 1 inverse pair.

Note:

The integer 

n

 is in the range [1, 1000] and 

k

 is
in the range [0, 1000].