Journal of Integer Sequences, Vol. 15 (2012), Article 12.1.1 |

Department of Mathematics

University of Haifa

31905 Haifa

Israel

Mark Shattuck

Department of Mathematics

University of Tennessee

Knoxville, TN 37996

USA

**Abstract:**

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.

(Concerned with sequence A000108.)

Received August 8 2011;
revised version received November 6 2011.
Published in *Journal of Integer Sequences*, December 15 2011.

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