For most permutation types of simplex-lattice hypergraphs, a new distance coloring produces more monochromatic cells than the greedy coloring, and it is optimal for a specific family.
Closed form formula for the number of restricted compositions
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Sperner's colorings of hypergraphs arising from edgewise triangulations
For most permutation types of simplex-lattice hypergraphs, a new distance coloring produces more monochromatic cells than the greedy coloring, and it is optimal for a specific family.