Counting peaks and valleys in $k$-colored Motzkin paths

Abstract

This paper deals with the enumeration of $k$-colored Motzkin paths with a fixed number of (left and right) peaks and valleys. Further enumeration results are obtained when peaks and valleys are counted at low and high level. Many well-known results for Dyck paths are obtained as special cases.

Enumerating Hamiltonian Cycles

“…The string representations of the border structures in S p G (U ) ordered by π are equivalent to 2-colored Motzkin words [17], which are strings with the character set {α 1 , α 2 , opening bracket '(', closing bracket ')'}, and the property that the brackets be properly nested. The string representation of a border structure can be mapped to the corresponding 2-colored Motzkin word by replacing endpoints of paths with opening and closing brackets, vertices forming a trivial path with α 1 , and vertices with degree two with α 2 .…”

“…Denote n = |B G (U )|. Now |S p G (U )| is the number of 2-colored Motzkin words of length n, which is known to be the (n + 1)th Catalan number C n+1 [17]. Similarly, |S g G (U )| is the number of Motzkin words of length n, i.e., the nth Motzkin number M n [1].…”

Dyck paths with a first return decomposition constrained by height

“…Restricted classes of Dyck paths have also been considered, for instance Barcucci et al [1] consider Dyck paths having a non-decreasing height sequence of valleys (see also [7,8]). Other papers deal with Motzkin paths using similar methods [2,5,10,11,16,21,24]. Motzkin and Catalan numbers appear alongside in many situations [10] and several one-to-one correspondences exist between restricted Dyck paths and Motzkin paths.…”

Hankel transforms of generalized Motzkin numbers

“…Motzkin numbers were generalized by introducing colorings of the paths. The t-colored Motzkin paths have horizontal (level) steps .1, 0/ colored by one of t colors [8,9]. The number of such paths is known as t-Motzkin number and denoted by m .t/ n .…”