  Stirling numbers, which count partitions of a set and permutations in the
symmetric group, have found extensive application in combinatorics, geometry,
and algebra. We study analogues and q-analogues of these numbers corresponding
to the Coxeter group of type B. In particular, we show how they are related to
complete homogeneous and elementary symmetric polynomials; demonstrate how they
q-count signed partitions and permutations; compute their ordinary,
exponential, and q-exponential generating functions; and prove various
identities about them. Ordered analogues of the q-Stirling numbers of the
second kind have recently appeared in conjectures of Zabrocki and of
Swanson--Wallach concerning the Hilbert series of certain super coinvariant
algebras. We provide conjectural bases for these algebras and show that they
have the correct Hilbert series.
