In the mathematical field of enumerative combinatorics, identities are sometimes established by arguments that rely on singling out one "distinguished element" of a set.
be a family of subsets of the set
be a distinguished element of set
Then suppose there is a predicate
that relates a subset
to be the set of subsets
to be the set of subsets
are disjoint sets, so by the method of summation, the cardinalities are additive[1] Thus the distinguished element allows for a decomposition according to a predicate that is a simple form of a divide and conquer algorithm.
In combinatorics, this allows for the construction of recurrence relations.
Examples are in the next section.