Distinct Initial Matrices

For a A * B matrix of all distinct numbers from 1 to A * B, we first sort each column and then concatenate all columns in increasing order of indices to form an array of size A * B. Columns are numbered in increasing order from left to right.

For example, if matrix is

[1 5 6]
[3 2 4]

We first sort all columns to get

[1 2 4]
[3 5 6]

Now, we concatenate columns in increasing order of indices to get an array

[1, 3, 2, 5, 4, 6]

Given this final array, you have to count how many distinct initial matrices are possible. Return the required answer modulo 10⁹+7.

Two matrices are distinct if:

• Either their dimensions are different.
• Or if any of the corresponding row in two matrices are different.

Example:

If input array is [1, 3, 2, 4], distinct initial matrices possible are:

[1 3 2 4]
-----------------------
[1 2]
[3 4]
-----------------------
[1 4]
[3 2]
-----------------------
[3 2]
[1 4]
-----------------------
[3 4]
[1 2]
-----------------------

that is, a total of 5 matrices.

NOTE: You only need to implement the given function. Do not read input, instead use the arguments to the function. Do not print the output, instead return values as specified. Still have a question? Checkout Sample Codes for more details.