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A function $$f : \{0,1\}^n \to \{0,1\}$$ is called symmetric if for every $x_1,x_2,\ldots,x_n \in \{0,1\}$ and every permutation $\sigma$ of $\{1,2,\ldots,n\}$, we have $$f(x_1,x_2,\ldots,x_n) = f(x_{\sigma(1)},x_{\sigma(2)},...,x_{\sigma(n)})$$ What is the number of such symmetric functions ?

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up vote 2 down vote accepted

If $f:\{0,1\}^n \rightarrow \{0,1\}$ is symmetric, the value of $f(x_1,x_2, \ldots x_n)$ will be determined simply by the number of 1's in $\{x_1, x_2 \ldots x_n\}$.

So partition $\{0,1\}^n$ into $n+1$ parts, say $\mathcal S_k$ each consisting $n$-bit strings with exactly $k$ number of 1's. The function $f$ maps an $n$-bit string to either 0 or 1. Hence the total number of such functions is $2^{n+1}$.

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