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

$\cal N$=2 SUSY and the Hexipentisteriruncicantitruncated 7-Simplex