Symmetric Boolean function

In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the permutation of its input bits, i.e., it depends only on the number of ones in the input.

From the definition follows, that there are 2ⁿ⁺¹ symmetric n-ary Boolean functions. It implies that instead of the truth table, traditionally used to represent Boolean functions, one may use a more compact representation for an n-variable symmetric Boolean function: the (n + 1)-vector, whose i-th entry (i = 0, ..., n) is the value of the function on an input vector with i ones.