Remember that one $ s $Partition is a partition of a natural number $ n $ so that everyone is part of the form $ 2 ^ r-1 $, According to a principle of Milnor is the number $ p_s (n) $ from $ s $-Partitions of $ n $ Counts the dimension of the Mod 2 Steenrod algebra in degrees $ n $, I am interested in the asymptotic function $ p_s (n) $and related functions for the odd-numbered primary Steenrod algebras.
Is the number of $ s $partitions $ p_s (n) $ grow sub-exponentially $ n $?
If so, there are effective constants $ p_s (n) leq C_ epsilon (1 + epsilon) ^ n $?
What about the dimension of the odd-numbered primary Steenrod algebras?
The OEIS page (here is the link again) leads to this article, in which an asymptotic formula is given for $ ln p_s (n) $and all expressions are indeed sublinear $ n $, except possibly for the notion of a crafting function $ W (z) $whose growth I can not estimate.
For the odd primary Steenrod algebras Milnor showed that z $ p $ an odd prime, the dual Steenrod algebra in prime $ p $ is the tensor product $ P ( xi_1, xi_2, dots) otimes E ( tau_0, tau_1, tau_2, dots) $ from where $ deg ( xi_i) = 2p ^ i – 2 $. $ deg ( tau_i = 2p ^ i – 1) $, and $ P, E $ denote polynomial and outer algebras, respectively $ mathbb F_p $, The counting of the dimension is thus reduced to a combinatorial partition problem with a similar taste.