A003183 - OEIS

%I M0814 #27 Oct 21 2023 06:34:58

%S 1,2,3,6,17,112,8282

%N Number of NPN-equivalence classes of unate Boolean functions of n or fewer variables.

%C Number of inequivalent (under the group of permutations and "inversion of variables") monotone Boolean functions of n of fewer variables.

%C Given f, a function of n variables, we define the "inversion of variables", i, by (i.f)(x1,...,xn)=1+f(1+x1,...,1+xn) (we can write (i.f)(x)=1+f(1+x) where the second "1" denotes (1,...,1)). It turns out that if f is monotone, then i.f is also monotone. On the other hand, a permutation of n elements, p, acts on f by (p.f)(x)=f(p(x)). It turns out that if f is monotone, then p.f is also monotone. We define p.i by (p.i)(f)=p.(i.f) and i.p by (i.p)(f)=i.(p.f). If we define a.b by (a.b).f=a.(b.f) for a,b elements of G, it turns out that G={p.i,p: p is a permutation of n elements} is a group. In this context, f and g are equivalent if there exists b (an element of G) such that b.f=g.

%D S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 18.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Aniruddha Biswas and Palash Sarkar, <a href="https://arxiv.org/abs/2304.14069">Counting unate and balanced monotone Boolean functions</a>, arXiv:2304.14069 [math.CO], 2023.

%H S. Muroga, <a href="/A000371/a000371.pdf">Threshold Logic and Its Applications</a>, Wiley, NY, 1971. [Annotated scans of a few pages]

%H <a href="/index/Bo#Boolean">Index entries for sequences related to Boolean functions</a>

%e a(2)=3 because m(x,y)=x, n(x,y)=y, k(x,y)=0, h(x,y)=1, f(x,y)=x*y, g(x,y)=x+y+xy are the six monotone Boolean functions of 2 or fewer variables; m and n are equivalent, k and h are equivalent, f and g are equivalent. Then we have 3 inequivalent monotone Boolean functions of 2 or fewer variables.

%Y Cf. A120608, A120587, A006602.

%K nonn,more

%O 0,2

%A _N. J. A. Sloane_

%E Additional comments from Alan Veliz-Cuba (alanavc(AT)vt.edu), Jun 18 2006