On Delannoy paths without peaks and valleys

Abstract

A lattice path is called Delannoy if each of its steps belong to {N,E,D}, where N=(0,1), E=(1,0), and D=(1,1) steps. A peak, a valley, and a deep valley are denoted by NE, EN, and EENN on the lattice path, respectively.Let Pn,m(NE,EN) be the set of Delannoy paths from the origin to (n,m) without peaks and valleys, and Pn,m(D,EENN) be the set of Delannoy lattice paths from the origin to (n,m) without diagonal steps and deep valleys. In this paper, we construct a bijection between Pn,m(NE,EN) and a specific subset of Pn,m(D,EENN). We also enumerate the number of Delannoy paths without peaks and valleys on the restricted region {(x,y)∈Z2:y≥kx} for a positive integer k.