On the Number of Equivalence Classes of Boolean and Invertible Boolean Functions

Abstract

The number <inline-formula> <tex-math notation= LaTeX >${U}_{n}$ </tex-math></inline-formula> of equivalence classes of Boolean functions of <italic>n</italic> variables and the number <inline-formula> <tex-math notation= LaTeX >${V}_{n}$ </tex-math></inline-formula> of equivalence classes of vectorial Boolean functions of <italic>n</italic> variables under the action of four groups of transformations are considered. The four groups are the group <inline-formula> <tex-math notation= LaTeX >${S}_{n} $ </tex-math></inline-formula> of permutations of variables, the group <inline-formula> <tex-math notation= LaTeX >${G}_{n}$ </tex-math></inline-formula> of permutations and complementations, the linear group <inline-formula> <tex-math notation= LaTeX >$\\mathrm {GL}({n},2)$ </tex-math></inline-formula> and the affine group <inline-formula> <tex-math notation= LaTeX >$\\mathrm {AGL}({n},2)$ </tex-math></inline-formula>. Harrison obtained cycle indexes for these groups and the expressions for <inline-formula> <tex-math notation= LaTeX >${U}_{n}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >${V}_{n}$ </tex-math></inline-formula> in terms of the corresponding cycle index. He also tabulated the numbers <inline-formula> <tex-math notation= LaTeX >${U}_{n}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation= LaTeX >${V}_{n}$ </tex-math></inline-formula> and the cycle indexes for <inline-formula> <tex-math notation= LaTeX >${n}\\le 6$ </tex-math></inline-formula> for <inline-formula> <tex-math notation= LaTeX >${S}_{n} $ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >${G}_{n}$ </tex-math></inline-formula>, and for <inline-formula> <tex-math notation= LaTeX >${n}\\le 5$ </tex-math></inline-formula> for <inline-formula> <tex-math notation= LaTeX >$\\mathrm {GL}({n},2)$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$\\mathrm {AGL}({n},2)$ </tex-math></inline-formula>. This bound was only recently slightly exceeded. Fripertinger implemented computation of cycle indexes for <inline-formula> <tex-math notation= LaTeX >$\\mathrm {GL}({n},{q})$ </tex-math></inline-formula> and <inline-formula> <tex-math notation= LaTeX >$\\mathrm {AGL}({n},{q})$ </tex-math></inline-formula>; if <inline-formula> <tex-math notation= LaTeX >${q}=2$ </tex-math></inline-formula> this implementation works for about <inline-formula> <tex-math notation= LaTeX >${n}\\le 21$ </tex-math></inline-formula>. By introducing appropriate precomputed tables, we reduced the cycle index computation to evaluation of a sum over partitions of n for all the four groups. Using this more efficient procedure, we obtained values of <inline-formula> <tex-math notation= LaTeX >${U}_{n}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation= LaTeX >${V}_{n}$ </tex-math></inline-formula> and the explicit cycle index expressions for larger values of <italic>n</italic>.