Dominance

The dominance relation on a set of points in Euclidean n-space is the intersection of the n coordinate-wise orderings. A point p dominates a point q provided that every coordinate of p is at least as large as the corresponding coordinate of q.

A partition p_a dominates a partition p_b if, for all k, the sum of the k largest parts of p_a is >= the sum of the k largest parts of p_b. For example, for n=7, {7} dominates all other partitions, while {1,1,1,1,1,1,1} is dominated by all others. In contrast, {3,1,1,1,1,} and {2,2,2,1} do not dominate each other (Skiena 1990, p. 52).

The dominance orders in R^n are precisely the partially ordered sets of dimension at most n.

See also

Connected Domination Number, Domatic Number, Domatic Partition, Dominating Set, Domination Number, Partially Ordered Set, Realizer

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References

Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, 1990.Stanton, D. and White, D. Constructive Combinatorics. New York: Springer-Verlag, 1986.

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Dominance

Cite this as:

Weisstein, Eric W. "Dominance." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Dominance.html

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