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A003183
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Number of NPN-equivalence classes of unate Boolean functions of n or fewer variables.
(Formerly M0814)
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0
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OFFSET
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0,2
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COMMENTS
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Number of inequivalent (under the group of permutations and "inversion of variables") monotone Boolean functions of n of fewer variables.
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.
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REFERENCES
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S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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Additional comments from Alan Veliz-Cuba (alanavc(AT)vt.edu), Jun 18 2006
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STATUS
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approved
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