RO-NMO 2005 Problem 113

Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$ , $n\geq 2$ . We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$ , where $a_{ij}=0$ , if $X_i \cap X_j = \emptyset$ , and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$ .