  We give multiple proofs of two formulas concerning the enumeration of
permutations avoiding a monotone consecutive pattern with a certain value for
the inverse peak number or inverse left peak number statistic. The enumeration
in both cases is given by a sequence related to Fibonacci numbers. We also show
that there is exactly one permutation whose inverse peak number is zero among
all permutations with any fixed descent composition, and we give a few
elementary consequences of this fact. Our proofs involve generating functions,
symmetric functions, regular expressions, and monomino-domino tilings.
