  We prove various formulas which express exponential generating functions
counting permutations by the peak number, valley number, double ascent number,
and double descent number statistics in terms of the exponential generating
function for Chebyshev polynomials, as well as cyclic analogues of these
formulas for derangements. We give several applications of these results,
including formulas for the $(-1)$-evaluation of some of these distributions.
Our proofs are combinatorial and involve the use of monomino-domino tilings,
the modified Foata-Strehl action (a.k.a. valley-hopping), and a cyclic analogue
of this action due to Sun and Wang.
