Some Enumerations on Non-Decreasing Dyck Paths

Abstract

We construct a formal power series on several variables that encodes many statistics on non-decreasing Dyck paths. In particular, we use this formal power series to count peaks, pyramid weights, and indexed sums of pyramid weights for all non-decreasing Dyck paths of length $2n.$ We also show that an indexed sum on pyramid weights depends only on the size and maximum element of the indexing set.

Enumerating symmetric peaks in non-decreasing Dyck paths

“…A Dyck path is called non-decreasing if the heights of its valleys form a non-decreasing sequence from left to right (see Figure 1 for an example). Non-decreasing Dyck paths have been extensively studied in the literature, see [2,5,6,8,14,16,17,20]. All the Dyck paths considered in this paper will be non-decreasing.…”

“…All the Dyck paths considered in this paper will be non-decreasing. Following the notation from [5,6,13,14], we denote by D the set of all non-decreasing Dyck paths, and by D n the set of all non-decreasing Dyck paths of length 2n, where the length is defined as the number of steps. For P ∈ D n , we write |P |= n to denote its semilength.…”

“…We are also interested in the size of the maximal pyramid containing a peak. We define the weight of a pyramid X h Y h to be equal to h. In [5,7], the authors refer to this parameter as the height, but we will use the term weight to suggest that it is not affected by the location of the pyramid. We define the weight of a peak to be the weight of the maximal pyramid that contains it.…”

Enumerations of Rational Non-decreasing Dyck Paths with Integer Slope

“…In 1997 Barcucci et al [2] introduced the concept of non-decreasing Dyck paths, and since then many papers, studying several aspects of this family of paths, have appeared [2,3,[6][7][8][9][10][11][12][13]19]. A t-Dyck path is a generalization of the Dyck path concept where every South-East step has length of ðt À 1Þ.…”

Dyck paths with a first return decomposition constrained by height

“…length, number of peaks, valleys, double rises and other pattern occurrences [9,14,15,17,18,19,20,23]. 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].…”