  We study algorithms for recursively creating arbitrary N-extended
`supermultiplets' given minimal matrix representations of off-shell, N = 1
supermultiplet matrices. We observe connections between the color vertex
problems in graph theory and the different supermultiplet sets locations in the
permutahedron by using the concepts of truncation and chromatic number. The
concept of `hopping operators' is introduced, constructed, and then used to
partition the 8! vertices of the permutahedron. We explicitly partition these
into 5,040 octets constrained in locations on the permutahedron by a magic
number rule. Boolean factors in this recursive construction are shown to obey a
doubly even binary flip rule. Although these hopping operators do not generally
constitute normal subgroups of the permutation group, we find that `ab-normal
cosets' exist where the same left- and right-hoppers appear as unordered sets.
Finally, using computer simulations, we investigate the types of faces on
higher-order permutahedron which may give rise to lower-order supermultiplets.
