We give an explicit formula for the number of n-variable clique function in terms of the parameters based upon the numbers of intersecting antichains of the lower half of the n-cube. We present the numbers of clique functions with up to seven variables through computer evaluation of the parameters.

We show that symmetric Boolean fanout-free functions or cascade functions realizable by networks of AND's, OR's and inverters are only the functions 0, 1, x1x2xn, x1+x2+xn, 12n, 1+2++n (n1). If we add exclusive OR gates as the elements of the networks, only the functions x1x2xn and x1x2xn+1 are realizable in addition to the above functions.

Equivalence classes of Boolean symmetric functions under negations and permutations of variables and negation of output are presented with their number.

A test for detection of functions that are in the same equivalence class under permutations and negations of variables and negation of output with a totally symmetric function based on spectral methods is presented. These functions have essentially identical circuit realizations as corresponding totally symmetric functions which gives significance of this detection.