  We prove that a beam splitter, one of the most common optical components,
fulfills several classes of majorization relations, which govern the amount of
quantum entanglement that it can generate. First, we show that the state
resulting from k photons impinging on a beam splitter majorizes the
corresponding state with any larger photon number k'>k, implying that the
entanglement monotonically grows with k. Then, we examine parametric
infinitesimal majorization relations as a function of the beam-splitter
transmittance, and find that there exists a parameter region where majorization
is again fulfilled, implying a monotonic increase of entanglement by moving
towards a balanced beam splitter. We also identify regions with a majorization
default, where the output states become incomparable. In this latter situation,
we find examples where catalysis may nevertheless be used in order to recover
majorization. The catalyst states can be as simple as a path-entangled
single-photon state or a two-mode vacuum squeezed state.
