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

A partition  dominates a partition  if, for all , the sum of the  largest parts of  is  the sum of the  largest parts of . For example, for , {7}" src="https://mathworld.wolfram.com/images/equations/Dominance/Inline16.gif" style="height:15px; width:17px" /> dominates all other partitions, while {1,1,1,1,1,1,1}" src="https://mathworld.wolfram.com/images/equations/Dominance/Inline17.gif" style="height:15px; width:107px" /> is dominated by all others. In contrast, {3,1,1,1,1,}" src="https://mathworld.wolfram.com/images/equations/Dominance/Inline18.gif" style="height:15px; width:81px" /> and {2,2,2,1}" src="https://mathworld.wolfram.com/images/equations/Dominance/Inline19.gif" style="height:15px; width:62px" /> do not dominate each other (Skiena 1990, p. 52).

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

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.