Non-uniform distribution with uniform subsets

Question

Let $X=[X_1, \dots, X_n]$ be a random vector of binary-valued random variables, taking values in $\{0,1\}^n$.

Let $S(n,k)$ denote the set of all size $k$ subsets of $\{1,\dots,n\}$.

Can anyone describe a joint distribution over $X$ such that each size $k$ subset $[X_{s(1)},\dots,X_{s(k)}], (s \in S(n,k), k=1,2,\dots,n-1)$ of the random vector $X$ is uniformly distributed over $\{0,1\}^{k}$, but $X$ is not uniformly distributed over $\{0,1\}^n$?

Answer 1

The uniform distribution over all vectors with an odd number of ones fits the bill.