  A recursion due to Kook expresses the Laplacian eigenvalues of a matroid M in
terms of the eigenvalues of its deletion M-e and contraction M/e by a fixed
element e, and an error term. We show that this error term is given simply by
the Laplacian eigenvalues of the pair (M-e, M/e). We further show that by
suitably generalizing deletion and contraction to arbitrary simplicial
complexes, the Laplacian eigenvalues of shifted simplicial complexes satisfy
this exact same recursion.
  We show that the class of simplicial complexes satisfying this recursion is
closed under a wide variety of natural operations, and that several
specializations of this recursion reduce to basic recursions for natural
invariants.
  We also find a simple formula for the Laplacian eigenvalues of an arbitrary
pair of shifted complexes in terms of a kind of generalized degree sequence.
