Let $\Omega\_n^O$ denote the abelian group of cobordism classes of closed, unoriented manifolds of dimension $n$, and let $d(n) := \lvert\Omega\_n^O\rvert$. What are the asymptotics of $d(n)$?


It's possible to express $\log\_2 d(n)$ as a modified partition function: by work of Thom, $\Omega\_n^O$ is an $\mathbb F\_2$-vector space with a basis of the form
$$\{x\_{\ell\_1,\dotsc,\ell\_j}\mid \ell\_1 + \dotsb + \ell\_j = n, \ell\_i \ne 2^r - 1\},\qquad\qquad (\*)$$
so $\dim\_{\mathbb F\_2}\Omega\_n^O = \log\_2 d(n)$ is equal to the number of partitions of $n$ into numbers that are not one less than a power of 2. Hence if $P(n)$ denotes the usual partition function, $d(n) = O(2^{P(n)})$. Is this sharp? What about lower bounds?



What is the growth rate of the number of unoriented cobordism classes?
If you denote by $P'(n)$ the number of partitions of $n$ with parts not of the form $2^k-1$, it holds that
$$\log P'(n)\sim \pi \sqrt{\frac{2n}{3}}$$
just like for the usual partition function. This will be true whenever you allow the parts to come from a set of integers of asymptotic density 1. See for example Nathanson's paper: ["Asymptotic density and the asymptotics of partition functions"](https://arxiv.org/abs/math/0002103).


