Laurent and power series over the field with one element? Dyckerhoff's paper arxiv.org/abs/1505.06940 says that finite $\mathbb F_1[[t]]$-modules should be considered as an $\mathbb F_1$ vector space (i.e. finite set with distinguished element $*$) together with a nilpotent endomorphism. (I guess this is a different paper from the one @darijgrinberg was looking at; the parts of the partition arise here as lengths of maximal paths to $*$.)