User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 2/@comment-7484840-20130530171941/@comment-7484840-20130530183526

italics means that that part of the array does not have to exist for the definition to work.

Which w(k)/ is worked on is described within the 2nd paragraph: it is the first one where the array after it has a non-1 entry before the 1st (k+1) separator. (could lead to it being the very 1st array in the chain).

The 'slightly altered version of @ (now @4)' refers to the changes that will be made by one of the rules for linear and multidimensional arrays before the nesting of the array (e.g reduce the active entry by one). The rest of the rule means that instead of just using @4 when nesting the function inside itself, you have to use @4 and all of the chain of w/'s that follows it.

as for the difference between @3 and @4, they are opposites of each other: when followed by w(k)/ in a w/ chain, @3 cannot have non-1 entries before the 1st (k+1) separator, and @4 must have non-1 entries before the 1st (k+1) separator.