2018 Putnam A2

Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some order, and let

$M$ be the $(2^n-1) \times (2^n-1)$ matrix whose $(i,j)$ entry is

$$

m_{ij} = \begin{cases} 0 & \text{if }S_i \cap S_j = \emptyset; \\

1 & \text{otherwise.}

\end{cases}

$$

Calculate the determinant of $M$.

设 $S_1, S_2, \dots, S_{2^n-1}$ 为 $\{1,2,\dots,n\}$ 按某种顺序的非空子集，并令

$M$ 是 $(2^n-1) \times (2^n-1)$ 矩阵，其 $(i,j)$ 条目为

$$

m_{ij} = \begin{cases} 0 & \text{if }S_i \cap S_j = \emptyset; \\

1 & \text{否则。}

\结束{案例}

$$

计算 $M$ 的行列式。