  Bernoulli-$p$ thinning has been well-studied for point processes. Here we
consider three other cases: (1) sequences $(X_1,X_2,...)$; (2) gaps of such
sequences $(X_{n+1}-X_1)_{n\in\mathbb{N}}$; (3) partition structures. For the
first case we characterize the distributions which are simultaneously invariant
under Bernoulli-$p$ thinning for all $p \in (0,1]$. Based on this, we make
conjectures for the latter two cases, and provide a potential approach for
proof. We explain the relation to spin glasses, which is complementary to
important previous work of Aizenman and Ruzmaikina, Arguin, and Shkolnikov.
