A refinement of Dyck paths: A combinatorial approach

Local maxima and minima of a Dyck path are called peaks and valleys, respectively. A Dyck path is called restricted[Formula: see text]-Dyck if the difference between any two consecutive valleys is at least [Formula: see text] (right-hand side minus left-hand side) or if it has at most one valley. In this paper, we use several techniques to enumerate some statistics over this new family of lattice paths. For instance, we use the symbolic method, the Chomsky–Schűtzenberger methodology, Zeilberger’s creative telescoping method, recurrence relations, and bijective relations. We count, for example, the number of paths of length [Formula: see text], the number of peaks, the number of valleys, the number of peaks of a fixed height, and the area under the paths. We also give a bijection between the restricted [Formula: see text]-Dyck paths and a family of binary words.

Enumerative Combinatorics of Intervals in the Dyck Pattern Poset

We initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

Maximality on Construction of Ternary Cross Bifix Free Code

The purpose of this research was to show that ternary cross bifix free code CBFS3(2m+1) and CBFS3(2m+2) achieved the maximum for every natural number m. This research was a literature review. A cross bifix free codes was constructed by using Dyck path method which achieved the maximality, that was non-expandable on binary set sequences for appropriate length. This result is obtained by partitioning members of CBFS3(2m+1) and CBFS3(2m+2) and comparing them with the maximality of CBFS2(2m+1) and CBFS2(2m+2). For small length 3, the result also shows that the code CBFS3(3) is optimal.

Short note on the number of 1-ascents in dispersed Dyck paths

A dispersed Dyck path (DDP) of length [Formula: see text] is a lattice path on [Formula: see text] from [Formula: see text] to [Formula: see text] in which the following steps are allowed: “up” [Formula: see text]; “down” [Formula: see text]; and “right” [Formula: see text]. An ascent in a DDP is an inclusion-wise maximal sequence of consecutive up steps. A 1-ascent is an ascent consisting of exactly 1 up step. We give a closed formula for the total number of 1-ascents in all dispersed Dyck paths of length [Formula: see text], #A191386 in Sloane’s OEIS. Previously, only implicit generating function relations and asymptotics were known.

KONSTRUKSI KODE CROSS BIFIX BEBAS TERNAIR BERPANJANG GENAP UNTUK MENGATASI MASALAH SINKRONISASI FRAME

In order to guarantee the synchronization between a transmited data by transmitter and received data by receiver can be done by periodically inserting a fixed sequence into the transmited data. It is one of the main topic in digital communication systems which called Frame Synchronization. Study of Cross Bifix Free Codes arise to solve Synchronization’s problem via distributed sequence’s method which introducted first in 2000. A Cross Bifix Free Codes is a set of sequences in which no prefix of any length of less than to of any sequences is the sufix of any sequence in the set. In 2012, a Binary Cross Bifix Free Codes was constructed by using Dyck path. In 2017, a Ternary Cross Bifix Free Codes with odd lenght was constructed, , by generalize the construction of binary cross bifix free. In this paper, will be constructed Ternary Cross Bifix Free Codes for even length, , by expand the construction of Binary Cross Bifix Free Codes.

KONSTRUKSI KODE CROSS BIFIX BEBAS TERNAIR BERPANJANG GENAP UNTUK MENGATASI MASALAH SINKRONISASI FRAME

In order to guarantee the synchronization between a transmited data by a transmitter and received data by the receiver can be done by periodically inserting a fixed sequence into the transmited data. It is one of the main topics in digital communication systems which called Frame Synchronization. Study of Cross Bifix Free Codes arises to solve Synchronization’s problem via distributed sequence’s method which introduced by Wigngardeen and Willink in 2000. A Cross Bifix Free Codes is a set of sequences in which no prefix of any length of less than to n of any sequences is the suffix of any sequence in the set. In 2012, Bilotta et al construct binary cross bifix free codes by using Dyck path. In 2017, Affaf et al was construct cross bifix free codes, 〖CBFS〗_3 (2m+1), by generalizing Bilotta’s construction. In this paper, will be constructed Ternary Cross Bifix Free Codes for even length, 〖CBFS〗_3 (2m+2), by using Bilotta and Affaf’s construction.(this paper has been submitted and still on review process in Jurnal Informatika dan Komputer (JIKO) at 16 August 2017)

Dinv and Area

We give a new combinatorial proof of the well known result that the dinv of an $(m,n)$-Dyck path is equal to the area of its sweep map image. The first proof of this remarkable identity for co-prime $(m,n)$ is due to Loehr and Warrington. There is also a second proof (in the co-prime case) due to Gorsky and Mazin and a third proof due to Mazin. A corrigendum was added to this paper on the 9th of June 2017.