This is a math heavy problem with little code.

Instead of problems and time, it may be easier to think about this as placing some blocks on a physical location in a line. We go through the blocks from 1 to n, and choose a number randomly, and try to place the block starting at the first empty position after the randomly chosen number. If this process succeeds for all blocks, then this corresponds to a valid scenario from the original problem. This process can fail if there isn't enough space for the block at the first empty position, or if there are no empty positions.

To solve this problem, there are a few observations. First, instead of placing blocks in a line with n empty spaces, let's place blocks in a circle with n+1 empty spaces. Blocks will scan from a position clockwise around the circle until they find an empty space to try to go in. This time, we are guaranteed every block will eventually find an empty space, but this process can still fail if there isn't enough space. In the end, we can multiply the answer by (# empty slots) / (n+1), since everything in on a circle, everything is symmetric, so the count of configurations where a particular space is empty we added is the same for all spaces.

So now, the main issue is counting the number of valid ways to place the blocks in a circle. As a first step, it doesn't matter where we put our first block since it's on a circle. As another simplifying assumption, let's assume there will only be one empty space at the end (we will talk about how to adjust the logic later to account for multiple empty spaces).

After we place this first block, we get a line, and let's partition this line into n sections (we don't fix the sizes of each sections yet, this will be determined by the order of the blocks after we're done). We will imagine these sections to exactly hold one block, and one section will be left empty.

For the second block, we choose a spot where it starts searching. There are two cases, this spot is occupied or empty. If the spot is empty, this means it must be the beginning of some section, so there are n ways to choose this. If the spot is occupied, then this second block will end up in a section that immediately follows an already occupied section. There are x_1 ways to choose this spot.

For the third block, we choose a spot where it starts searching. Again, there are two cases, this spot is occupied or empty. If this spot is empty, this is the beginning of some section, of which there are n-1 ways to choose this. If the spot is occupied, then the third block will end up in a section that immediately follows an already occupied section. There are x_1 + x_2 ways to choose this spot (and notice, this is always the same regardless of where the first and second blocks are adjacent or not).

Generalizing, we can see the i-th block has (n - i + 2 + x_1 + x_2 + ... + x_i) choices of a spot. Thus, the answer is just the product of all these choices.

To deal with empty spaces, instead of starting with n sections, we start with (n-1+#empty slots) sections, and the resulting logic is similar.

To summarize, the main observations are:
- Do this on a circle instead of a line
- Split the circle into sections, but don't fix the sizes of the sections while the process is going. The section sizes will be determined after the order of the blocks is determined
