Design of crystal-like lattices can be achieved by using some net operations. Hypothetical networks, thus obtained, can be characterized in their topology by various counting polynomials and topological indices derived from them. The networks herein presented are related to the Dyck graph and described in terms of Omega polynomial and PIv polynomials.

In this paper we study the class of generalized Motzkin paths with no hills and prove some of their combinatorial properties in a bijective way; as a particular case we have the Fine numbers, enumerating Dyck paths with no hills. Using the ECO method, we define a recursive construction for Dyck paths such that the number of local expansions performed on each path depends on the number of its hi...

Among the most familiar systems in symbolic dynamics are the subshifts of finite type, or SFT’s for short. SFT’s are relatively easy to analyze, and have many pleasant properties such as intrinsic ergodicity and unique ergodicity with respect to the tail. A larger class, which has the desirable property of being closed under factors, is that of sofic systems. This class of systems retains many ...

A Dyck shift and a Motzkin shift are mathematical models for constraints on genetic sequences. In terms of the theory of symbolic dynamics, neither of the Dyck shift nor the Motzkin shift is sofic. In terms of the mathematical language theory, they are non-regular and context free languages. Therefore we can not use the Perron-Frobenius theory to calculate capacities of these constraints. O. Mi...

For any given k, the sequence of k-ary Catalan numbers, Ct,k = 1 kt+1 ( kt t ) , enumerates a number of combinatorial objects, including k-ary Dyck words of length n = kt and k-ary trees with t internal nodes. We show that these objects can be efficiently ordered using the same variation of lexicographic order known as cool-lex order. In particular, we provide loopless algorithms that generate ...

The fundamental problem of enumerative combinatorics is to determine the number of elements of a set. More precisely, given an infinite indexed collection {Ai}i∈I of finite sets, we want to find a formula for the cardinality of Ai as a function of i, or at least a method for determining the number of elements of Ai that is easier (or more interesting) than counting them one at a time. There are...

New topological operations are introduced in order to recover the generalized Dyck equations presented by D. Arquès et al. in another way for the generating functions of maps and colored maps, by decomposing maps topologically and bijectively. By repeatedly applying the operations which made it possible to reveal the generalized Dyck equations for the successive transformed maps, oneto-one corr...

There are at least three di erent bijections in the literature from Dyck paths to 321-avoiding permutations, due to Billey-Jockusch-Stanley, Krattenthaler, and Mansour-Deng-Du. How di erent are they? Denoting them B;K;M respectively, we show that M = B Æ L = K Æ L where L is the classical Kreweras-Lalanne involution on Dyck paths and L0, also an involution, is a sort of derivative of L. Thus K ...

Recently, Kenyon and Wilson introduced a certain matrix M in order to compute pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix M−1 is equal to the number of certain Dyck tilings of a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of M−1. In this...