  We present Binomials, a package for the computer algebra system Macaulay2,
which specializes well known algorithms to binomial ideals. These come up
frequently in algebraic statistics and commutative algebra, and it is shown
that significant speedup of computations like primary decomposition is
possible. While central parts of the implemented algorithms go back to Eisenbud
and Sturmfels (1996), we also discuss a new algorithm for computing the minimal
primes of a binomial ideal. All decompositions make significant use of
combinatorial structure found in binomial ideals, and to demonstrate the power
of this approach we show how Binomials was used to compute primary
decompositions of commuting birth and death ideals of Evans et al., yielding a
counterexample for a conjecture therein.
