  In a uniform random permutation \Pi of [n] := {1,2,...,n}, the set of
elements k in [n-1] such that \Pi(k+1) = \Pi(k) + 1 has the same distribution
as the set of fixed points of \Pi that lie in [n-1]. We give three different
proofs of this fact using, respectively, an enumeration relying on the
inclusion-exclusion principle, the introduction of two different Markov chains
to generate uniform random permutations, and the construction of a
combinatorial bijection. We also obtain the distribution of the analogous set
for circular permutations that consists of those k in [n] such that \Pi(k+1 mod
n) = \Pi(k) + 1 mod n. This latter random set is just the set of fixed points
of the commutator [\rho, \Pi], where \rho is the n-cycle (1,2,...,n). We show
for a general permutation \eta that, under weak conditions on the number of
fixed points and 2-cycles of \eta, the total variation distance between the
distribution of the number of fixed points of [\eta,\Pi] and a Poisson
distribution with expected value 1 is small when n is large.
