In combinatorics, double counting, also called counting in two ways, is a combinatorial proof technique for showing that two expressions are equal by demonstrating that they are two ways of counting the size of one set. In this technique, which van Lint & Wilson (2001) call “one of the most important tools in combinatorics,” one describes a finite set X from two perspectives leading to two distinct expressions for the size of the set. Since both expressions equal the size of the same set, they equal each other.
Famous quotes containing the words double and/or counting:
“One key, one solution to the mysteries of the human condition, one solution to the old knots of fate, freedom, and foreknowledge, exists, the propounding, namely, of the double consciousness. A man must ride alternately on the horses of his private and public nature, as the equestrians in the circus throw themselves nimbly from horse to horse, or plant one foot on the back of one, and the other foot on the back of the other.”
—Ralph Waldo Emerson (1803–1882)
“Love is sinister,
is mean to us in separation;
makes our thin bodies thinner.
This fellow Death
lacks mercy
and is good at counting our days.
And Master,
you, too, are subject
to the plague of jealousy
so think:
how could womenfolk,
soft as sprouts,
live like this?”
—Amaru (c. seventh century A.D.)