This page is for seriesmovers. That is to say that one player makes all (or most of) the moves. For example, in a series (direct) mate, (denoted s#n) White will make n moves ending in mate without Black making a single move. In a series help mate (sh#n), Black makes n moves and then White makes a single, mating, move. It's an additional constraint that there should be no intermediate checks, except possibly for Black's final move in a sh#n.
The problem shown here are consequent seriesmovers (cs#n, csh#n). These are my favourite kind. In a consequent seriesmover, any intermediate position must be legal with the appropriate person to move. This directly implies, for example, the non-checking constraint mentioned earlier. This means also that retro-analysis can be used to deduce the legality of castling or en passant, with the default conventions applying.
cs#n (for n>1) means White to play and produce a cs#(n-1).
csh#n (for n>1) means Black to play and produce a csh#(n-1).