This notation uses the same rules as the Sbiis Saibian's Hyper-E notation, and the rules are as follows:
If there are no hyperions: F[b]x = b^x (If there is only one argument x, the value of the expression is b^x.)
If the last entry is 1: F[b]a(1)#a(2)#a(3)#...#a(n)#1 = F[b]a(1)#a(2)#a(3)#...#a(n) (If the last entry is 1, it may be removed.)
Otherwise:
F[b]a(1)#a(2)#a(3)#...#a(n-2)#a(n-1)#a(n) =
F[b]a(1)#a(2)#a(3)#...#a(n-2)#F[b]a(1)#a(2)#a(3)#...#a(n-2)#a(n-1)#(a(n) - 1)
Evaluate the original expression, but with the last entry decreased by 1. Call this value z.
Remove the last two entries of the expression.
Add z as an entry to the end of the expression.
The priority of the rules are stated in the Saibian's webpage below the definition, based on "There are 3 rules used. If your given a E# expression, you first check to see if rule 1 applies. If it does not you check for rule 2. If that doesn't apply you go to 3." It disables a(n) being 1 in the 3rd rule and avoids an overlapping case classification. The Pre-Hyper-F notation has the same behavior to that of the basic Hyper-E notation.
Extended Hyper-F notation