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A102449 Number of canalizing Boolean functions with n input variables. 3
2, 4, 14, 120, 3514, 1292276, 103071426294, 516508833342349371376, 10889035741470030826695916769153787968498 (list; graph; refs; listen; history; text; internal format)



A canalizing function is one in which at least one of the input variables is able to determine the value of the output of the function regardless of the other variables. For example, f(x1,x2,x3) = x1 + x2*x3, where + is disjunction and * is conjunction, is a canalizing function, since setting x1 = 1 guarantees that the function is 1 regardless of the value of x2 or x3. On the other hand, the function f(x1, x2) = x1 + x2, where + is addition modulo 2, is not a canalizing function, since the values of both variables always need to be known in order to determine the function output.


D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.


Table of n, a(n) for n=0..8.

W. Just, I. Shmulevich and J. Konvalina, The number and probability of canalizing functions, arXiv:math-ph/0312033, 2003; Physica D, Vol. 197, pp. 211-221, 2004.

I. Shmulevich, H. Laehdesmaeki, E. R. Dougherty, J. Astola and W. Zhang, The role of certain Post classes in Boolean network models of genetic networks, Proceedings of the National Academy of Sciences of the USA, Vol. 100, No. 19, pp. 10734-10739, 2003.


a(n) = 2((-1)^n - n) + Sum_{k=1..n} (-1)^(k+1)*binomial(n, k)*2^(k+1)*2^(2^(n-k)).


Cf. A109460, A000616, A109462.

Sequence in context: A329234 A238638 A240973 * A193520 A102897 A305856

Adjacent sequences:  A102446 A102447 A102448 * A102450 A102451 A102452




Ilya Shmulevich (is(AT)ieee.org), Feb 23 2005



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