Enumerating symmetric and asymmetric peaks in Dyck paths

Enumerating symmetric peaks in non-decreasing Dyck paths

“…Flórez and Ramírez [15] introduced the concept of symmetric and asymmetric peaks in Dyck paths, see also recent follow-up work by Elizalde [11] and Flórez et al [13]. This concept was motivated in part by Asakly's [1] study of symmetric and asymmetric peaks in k-ary words.…”

Symmetric and asymmetric peaks or valleys in (partial) Dyck paths

“…See Figure 1 for detailed illustrations. In the literature, there are many papers dedicated to statistics of Dyck paths (words), see [1,2,3,4,5,6,7,8,9,10,11], [15,16] and the references therein. Recently, Flóres and Ramírez [8] find a formula for the total number, sp(n), of symmetric peaks over all Dyck paths of length 2(n + 1), as well as for the total number, ap(n), of asymmetric peaks over all Dyck paths of length 2(n + 3).…”

“…In the literature, there are many papers dedicated to statistics of Dyck paths (words), see [1,2,3,4,5,6,7,8,9,10,11], [15,16] and the references therein. Recently, Flóres and Ramírez [8] find a formula for the total number, sp(n), of symmetric peaks over all Dyck paths of length 2(n + 1), as well as for the total number, ap(n), of asymmetric peaks over all Dyck paths of length 2(n + 3). Elizalde [6] obtains a trivariate generating function that enumerates Dyck paths with respect to the number of symmetric peaks and the number of asymmetric peaks.…”

Symmetric peaks and symmetric valleys in Dyck paths

“…There is an extensive literature on the enumeration of Dyck paths with respect to the number of occurrences of certain subpaths and other related statistics, see for example [6,8,15,1]. In this paper we explore a different type of statistic that has been recently introduced by Flórez and Rodríguez [12], namely the number of symmetric and asymmetric peaks, as well as a new related statistic that we call the number of symmetric valleys. We remark that this notion of symmetry is unrelated to the degree of symmetry statistic defined in [10].…”

“…Throughout the paper, occurrences of subpaths always refer to subsequences in consecutive positions. In [12], Flórez and Rodríguez introduce the notion of symmetric peaks, which can be defined as follows. First, note that every peak can be extended to a unique maximal subsequence of the form u i d j for i, j ≥ 1, which we call the maximal mountain of the peak.…”