  A subtraction-free definition of the big Witt vector construction was
recently given by the first author. This allows one to define the big Witt
vectors of any semiring. Here we give an explicit combinatorial description of
the big Witt vectors of the Boolean semiring. We do the same for two variants
of the big Witt vector construction: the Schur Witt vectors and the $p$-typical
Witt vectors. We use this to give an explicit description of the Schur Witt
vectors of the natural numbers, which can be viewed as the classification of
totally positive power series with integral coefficients, first obtained by
Davydov. We also determine the cardinalities of some Witt vector algebras with
entries in various concrete arithmetic semirings.
