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. In this paper we present an algorithm that generates all words of length 2n of the generalized Dyck language lexicograph-ically. Thereby, each Dyck wo...

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...

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...

We exploit Krattenthaler’s bijection between 123-avoiding permutations and Dyck paths to determine the Eulerian distribution over the set Sn(123) of 123-avoiding permutations in Sn. In particular, we show that the descents of a permutation correspond to valleys and triple ascents of the associated Dyck path. We get the Eulerian numbers of Sn(123) by studying the joint distribution of these two ...

We give some interpretations to certain integer sequences in terms of parameters on Grand-Dyck paths and coloured noncrossing partitions, and we find some new bijections relating Grand-Dyck paths and signed pattern avoiding permutations. Next we transfer a natural distributive lattice structure on Grand-Dyck paths to coloured noncrossing partitions and signed pattern avoiding permutations, thus...

Kim and Drake used generating functions to prove that the number of 2-distant noncrossing matchings, which are in bijection with little Schröder paths, is the same as the weight of Dyck paths in which downsteps from even height have weight 2. This work presents bijections from those Dyck paths to little Schröder paths, and from a similar set of Dyck paths to big Schröder paths. We show the effe...

The distribution of the (absolute) area of Brownian motion was first studied by Mark Kac [Kac (1949)], using a formula since known as the Feynman–Kac formula [Kac (1949); Ito and McKean (1965)], which allows us to find the double Laplace transform of the distribution of the integral of a functional of Brownian motion. The area functional has received intensive attention for different types of B...

Dyck tilings were introduced by Kenyon and Wilson in their study of double-dimer pairings. They are certain kinds of tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths. We give two bijections between “cover-inclusive” Dyck tilings and linear extensions of tree posets. The first bijection maps the statistic (area + tiles)/2 to inversions of the linear extension, and the seco...