A switching function of n variables assigns to each n-digit binary sequence a value of 0 or 1...?

A switching function of n variables assigns to each n-digit binary sequence a value of 0 or 1. A switching function is called self-dual if the assigned value of a sequence S is unchanged when the 0 digits of S are changed to 1 and vice versa. How many self-dual switching functions of n variables are there?|||I'm afraid we'll have to assume "assigned value" means "total sum of the digits" or something.





Otherwise your explanation is too vague to figure out.








If v(S) = v(S' ), where S' denotes the dual of S, then you have:





v(S) = sum(S) = # of 1s in S





v(S' ) = sum(S' ) = # of 1s in S' = # of 0s in S





Thus S is self-dual if it has the same number of 0s as 1s.





For n odd, this is impossible. For n = 2k, the answer is (2k choose k). Refer to the other question of yours that I answered for why/how "choose" is defined and relevant to this question.