Organize the interpretations of various combinatorial objects
The factoring (number of simplexes n choose k - dependent simplex) x (number of flags on k - independent Euclidean) x (number of flags on n-k - independent Euclidean) = (number of flags on n)
The combinatorial interpretation of n-choose-k counts placements = "external arrangements" n! x...x (n-k+1)! and then divides by the redundancies = "internal arrangements" k! Thus it relates external and internal (within subsystem).
Conjugation gives the ways of relabeling, renaming. For example, (132)(12)(123) relables 1 as 2 and 2 as 3 in (12) to get (23).