Let \(n,m\ge 0\), \(\pi \colon [n]\to [m]\), \(j\in [m]\), and \(i\in [n]\). For \(F_j(\pi )=\{ u\in [n]:\pi (u)=j\} \), one has \(i\in F_j(\pi )\) if and only if \(\pi (i)=j\).

For \(n\ge 0\) and \(x\in \{ 0,1\} ^n\), flipping all coordinates gives the vector \(\bar x\) with \(\bar x_i=1-x_i\) for every \(i\in [n]\).

Let \(n\ge 0\), \(x\in \{ 0,1\} ^n\), and \(S\subseteq [n]\), and write \(x^S\) for the vector formed by flipping exactly the bits in \(S\). Then \((x^S)^S=x\).

For \(n\ge 0\) and \(z,c\in \{ 0,1\} ^n\), let \(\oplus \) denote coordinatewise addition modulo two. Then \((z\oplus c)\oplus c=z\).

Let \(n\ge 0\), \(F\colon \{ 0,1\} ^n\to \mathbb R\), and \(S\subseteq [n]\); write \(x^S\) for the vector formed by flipping the coordinates in \(S\). Then \(\sum _{x\in \{ 0,1\} ^n}F(x^S)=\sum _{x\in \{ 0,1\} ^n}F(x)\).

Let \(n\ge 0\), \(F\colon \{ 0,1\} ^n\to \mathbb R\), and \(c\in \{ 0,1\} ^n\), with \(\oplus \) denoting coordinatewise addition modulo two. Then \(\sum _{z\in \{ 0,1\} ^n}F(z\oplus c)=\sum _{x\in \{ 0,1\} ^n}F(x)\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). For \(S\subseteq [n]\), let \(x^S\) be the bit vector obtained by flipping exactly the coordinates in \(S\). For all \(S,T\subseteq [n]\) and \(x\in \{ 0,1\} ^n\), \(\chi _T(x^S)=(-1)^{|T\cap S|}\chi _T(x)\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). For \(S\subseteq [n]\), let \(x^S\) be the bit vector obtained by flipping exactly the coordinates in \(S\). For every real cube function \(f\) and sets \(S,T\subseteq [n]\), \(\widehat{(x\mapsto f(x^S))}(T)=(-1)^{|T\cap S|}\widehat f(T)\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). For \(S\subseteq [n]\), let \(x^S\) be the bit vector obtained by flipping exactly the coordinates in \(S\). For every real cube function \(f\) and \(S\subseteq [n]\), \(\mathbb E[f(x)f(x^S)]=\sum _{T\subseteq [n]}(-1)^{|T\cap S|}\widehat f(T)^2\).

Let \(n\ge 0\) and equip \(\{ 0,1\} ^n\) with its uniform probability measure; write \(\mathbb E\) for uniform expectation. For \(S\subseteq [n]\), put \(\chi _S(x)=\prod _{i\in S}(-1)^{x_i}\) and \(\widehat h(S)=\mathbb E[h\chi _S]\) for real cube functions \(h\). For \(S\subseteq [n]\), let \(x^S\) be the bit vector obtained by flipping exactly the coordinates in \(S\). If \(f\colon \{ 0,1\} ^n\to \{ -1,1\} \), then \(\Pr [f(x)\ne f(x^S)]=(1-\sum _{T\subseteq [n]}(-1)^{|T\cap S|}\widehat f(T)^2)/2\) for each \(S\subseteq [n]\); equivalently, this is \(\tfrac 14\mathbb E[(f(x)-f(x^S))^2]\).

Let \(n,m\ge 0\), \(j_0\in [m]\), \(T\subseteq [n]\), and \(\pi \colon [n]\to [m]\), and put \(F_{j_0}(\pi )=\{ i:\pi (i)=j_0\} \). Then \((-1)^{|T\cap F_{j_0}(\pi )|}=\prod _{i=1}^n b_i\), where \(b_i=-1\) if \(i\in T\) and \(\pi (i)=j_0\), and \(b_i=1\) otherwise.

Let \(n\ge 0\), \(m\gt 0\) be integers, \(j_0\in [m]\), and \(T\subseteq [n]\). For \(F_{j_0}(\pi )=\{ i\in [n]:\pi (i)=j_0\} \), the sum over all colorings satisfies \(\sum _{\pi :[n]\to [m]}(-1)^{|T\cap F_{j_0}(\pi )|}=m^n(1-2/m)^{|T|}\).