The class of normal subshifts includes irreducible nontrivial topological Markov shifts, sofic synchronized systems, Dyck β-shifts, substitution minimal and so on. We will characterize one-sided conjugacy classes in terms the associated C∗-algebras its gauge actions with potentials. In particular, shifts are characterized simple purely infinite Cuntz-Krieger algebras matrices left Fischer cover...

We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path P , we determine a formula for the number of Dyck paths covered by P , as well as for the ...

In this note we examine the number of integer lattice paths consisting of up-steps (1, 1) and down-steps (1,−1) that do not touch the lines y = m and y = −k, and in particular Theorem 3.2 in [P. Mladenović, Combinatorics, Mathematical Society of Serbia, Belgrade, 2001]. The theorem is shown to be incorrect for n > m + k + min(m, k), and using similar combinatorial technique we proved the upper ...

We study some basic properties of sofic-Dyck shifts and finite-type-Dyck shifts. We prove that the class of sofic-Dyck shifts is stable under proper conjugacies. We prove a Decomposition Theorem of a proper conjugacy between edge-Dyck shifts into a sequence of Dyck splittings and amalgamations.

Dynamic complexity is concerned with updating the output of a problem when the input is slightly changed. We study the dynamic complexity of two-player parity games over graphs of bounded tree-width, where updates may add or delete edges, or change the owner or color of states. We show that this problem is in DynFO (with LOGSPACE precomputation); this is achieved by a reduction to a Dyck-path p...

We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path P , we determine a formula for the number of Dyck paths covered by P , as well as for the ...

Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths. The enumeration of Dyck tilings is related with hook formulas for forests and the combinatorics of Hermite polynomials. The first goal of this work is to give an alternative point of view on Dyck tilings by making use of the weak order and the Bruhat order on permutations. Then w...

Given two disjoint alphabets T and T ] and a relation R T T ] , the generalized Dyck language D R over T T ] consists of all words w 2 (T T ]) ? which are equivalent to the empty word " under the congruence deened by x y " mod for all (x; y) 2 R. If the Dyck words are arranged according to the lexicographical order, then ranking means to determine the rank, i. e. the position, of a Dyck word. U...

Let τ be a fixed lattice path (called in this context string) on the integer plane, consisting of two kinds of steps. The Dyck path statistic “number of occurrences of τ” has been studied by many authors, for particular strings only. In this paper, arbitrary strings are considered. The associated generating function is evaluated when τ is a Dyck prefix (or a Dyck suffix). Furthermore, the case ...