A
Dyck path of length 2
n is a lattice path from (0,0) to
(2
n,0) consisting of up-steps
u=(1,1) and down-steps
d=(1,-1)which never passes below the
x-axis. Let
denote the set of
Dyck paths of length 2
n. A
peak is an occurrence of
ud (an
upstep immediately followed by a downstep) within a Dyck path, while a
valley is an occurrence of
du. Here, we compute explicit
formulas for the generating functions which count the members of
according to the maximum number of steps between any two peaks,
any two valleys, or a peak and a valley. In addition, we provide closed
expressions for the total value of the corresponding statistics taken
over all of the members of
.
Equivalent statistics on the set
of 231-avoiding permutations of length
n are also described.