Counting strings in Dyck paths

Some strings in Dyck paths

Counting Generalized Dyck Paths

The Catalan number has a lot of interpretations and one of them is the number of Dyck paths. A Dyck path is a lattice path from (0, 0) to (n, n) which is below the diagonal line y = x. One way to generalize the definition of Dyck path is to change the end point of Dyck path, i.e. we define (generalized) Dyck path to be a lattice path from (0, 0) to (m, n) ∈ N2 which is below the diagonal line y...

Counting Upper Interactions in Dyck Paths

A Dyck word w is a word over the alphabet {x, x} that contains as many letters x as letters x and such that any prefix contains at least as many letters x as letters x. The size of w is the number of letters x in w. A Dyck path is a walk in the plane, that starts from the origin, is made up of rises, i.e. steps (1, 1), and falls, i.e. steps (1,−1), remains above the horizontal axis and finishes...

Counting Segmented Permutations Using Bicoloured Dyck Paths

A bicoloured Dyck path is a Dyck path in which each up-step is assigned one of two colours, say, red and green. We say that a permutation π is σ-segmented if every occurrence o of σ in π is a segment-occurrence (i.e., o is a contiguous subword in π). We show combinatorially the following results: The 132-segmented permutations of length n with k occurrences of 132 are in one-to-one corresponden...

General Results on the Enumeration of Strings in Dyck Paths

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 ...