Priority Customers (Hard)

**This is the hard version of the problem. Here, you need to compute the number of valid subsets of each size. The constraints are the same as the easy version.** $N$ people stand in line to enter your store. Some of them are priority customers, though you don't know which ones yet. Their initial order is $[1, 2, 3, \ldots, N]$. You can change the order of people by doing the following: Call