  We study a class of random polymers, introduced by Sina\u{i}, which are
related to persistence probabilities in integrated simple random walk bridges.
We find the precise asymptotics of these probabilities, and describe their
combinatorics, using limit theory for infinitely divisible distributions, and
the number-theoretic subset counting formulas of von Sterneck from the early
1900s. Our results sharpen estimates by Aurzada, Dereich and Lifshits, and
respond to a conjecture of Caravenna and Deuschel, which arose in their study
of the pinning/wetting models, for random linear chains exhibiting entropic
repulsion. Our key combinatorial result is an analogue of Sparre Andersen's
classical formula.
