  We use the notion of Borel generators to give alternative methods for
computing standard invariants, such as associated primes, Hilbert series, and
Betti numbers, of Borel ideals. Because there are generally few Borel
generators relative to ordinary generators, this enables one to do manual
computations much more easily. Moreover, this perspective allows us to find new
connections to combinatorics involving Catalan numbers and their
generalizations. We conclude with a surprising result relating the Betti
numbers of certain principal Borel ideals to the number of pointed
pseudo-triangulations of particular planar point sets.

Commutative Algebra;Combinatorics