Proposition 2

Let $A$ be a ring, $x_1, \ldots, x_n$ elements of $A$ and $I = \{1, 2, \ldots, n\}$. For $H \subset I$, we write $x_H = \sum_{i \in H} x_i$. Then

*In particular, if $A$ is commutative,

Let $C$ be the set of mappings of $I$ into $\{0, 1\}$. If each $H \subset I$ is mapped to its characteristic function, a bijection is obtained of $\mathscr{P}(I)$ onto $C$. The right hand side of (16) is thus equal to:

where

(1) Suppose that $(i_1, \ldots, i_n)$ is not a permutation of $I$. There exists a $j \in I$ distinct from $i_1, \ldots, i_n$. Let $C'$ be the set of $a \in C$ such that $a(j) = 0$. For all $a \in C'$, let $a^*$ be the sum of $a$ and the characteristic function of $\{j\}$. Then $a^*(1) + \cdots + a^*(n) = a(1) + \cdots + a(n) + 1$ and hence

(2) Suppose that there exists $\sigma \in S_n$ such that $i_1 = \sigma(1), \ldots, i_n = \sigma(n)$. Then $a(i_1) \ldots a(i_n) = 0$ unless $a$ only takes the value 1. Thus $c_{i_1 \ldots i_n} = (-1)^n$.