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.

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.