Counting Dyck Paths According to the Maximum Distance Between Peaks and Valleys
نویسندگان
چکیده
A Dyck path of length 2n is a lattice path from (0, 0) to (2n, 0) consisting of upsteps u = (1, 1) and down-steps d = (1,−1) which never passes below the x-axis. Let Dn denote the set of Dyck paths of length 2n. 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 Dn 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 Dn. Equivalent statistics on the set of 231-avoiding permutations of length n are also described.
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