The group equation abcdef = dabecf can be reduced to the equation xcde = dxec. In general, we are interested in how many variables are needed to represent group equations in which the right side is a permutation of the variables on the left side. Scrambling numbers capture this information about a permutation. In this paper we present several facts about scrambling numbers, and expose a striking relationship between permutations that cannot be reduced and derangements.
Balof, Barry; Farmer, Eric; and Kawabata, Jamie, "The Link between Scrambling Numbers and Derangements" (1997). Mathematical Sciences Technical Reports (MSTR). 70.