Recall that an $s$-partition is a partition of a natural number $n$ such that each part is of the form $2^r-1$. By a fundamental theorem of Milnor, the number $p_s(n)$ of $s$-partitions of $n$ counts the dimension of the mod-2 Steenrod algebra in degree $n$. I'm interested in the asymptotics of the function $p_s(n)$, as well as related functions for the odd-primary Steenrod algebras.

**Questions:**

Does the number of $s$-partitions $p_s(n)$ grow subexponentially in $n$?

If so, are there effective constants $p_s(n) \leq C_\epsilon (1+\epsilon)^n$?

What about the dimension of the odd-primary Steenrod algebras?

The OEIS page (here's the link again) leads to this paper which gives an asymptotic formula for $\ln p_s(n)$, and all the terms are indeed sublinear in $n$, except possibly for the term involving a handicrafted function $W(z)$, whose growth I don't know how to estimate.

As for the odd-primary Steenrod algebras, Milnor showed that for $p$ an odd prime, the dual Steenrod algebra at the prime $p$ is the tensor product $P(\xi_1,\xi_2,\dots) \otimes E(\tau_0,\tau_1,\tau_2,\dots)$ where $deg(\xi_i) = 2p^i - 2$, $deg(\tau_i = 2p^i - 1)$, and $P, E$ denote polynomial and exterior algebras respectively, over $\mathbb F_p$. So counting the dimension reduces to a combinatorial partition problem of a similar flavor.