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For integers k ≥ 1 and n ≥ 2k + 1, the Kneser graph K(n, k) is the graph whose vertices are the k-element subsets of {1, . . . , n} and whose edges connect pairs of subsets that are disjoint. The Kneser graphs of the form K(2k + 1, k) are also known as the odd graphs. We settle an old problem due to Meredith, Lloyd, and Biggs from the 1970s, proving that for every k ≥ 3, the odd graph K(2k + 1,...

In this note, we present constructive bijections from Dyck and Motzkin meanders with catastrophes to paths avoiding some patterns. As a byproduct, deduce correspondences excursions restricted paths.

Indexed languages are languages recognized by pushdown automata of level 2 and by indexed grammars. We propose here some new characterizations linking indexed languages to context-free languages: the class of indexed languages is the image of the Dyck language by a nice class of context-free transducers, it is also the class of images, by a projection defined by an FO-formula of nested words la...

In this paper, we focus on Dyck paths with peaks and valleys, avoiding an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak and valley heights to avoid congruence classes modulo k. We study the shift equivalence on sequences, which in turn induces an equivalence relation...

In this paper we introduce a new bijection from the set of Dyck paths to itself. This bijection has the property that it maps statistics that appeared recently in the study of pattern-avoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. We also present a generalization of the bijection, as well as several applications of it to enumeration problems...

We construct a formal power series on several variables that encodes many statistics on non-decreasing Dyck paths. In particular, we use this formal power series to count peaks, pyramid weights, and indexed sums of pyramid weights for all nondecreasing Dyck paths of length 2n. We also show that an indexed sum on pyramid weights depends only on the size and maximum element of the indexing set.

In the present paper, we consider two kinds of statistics “number of usegments” and “number of internal u-segments” in Dyck paths. More precisely, using Lagrange inversion formula we present the generating function for the number of Dyck paths according to semilength and our new statistics by the partial Bell polynomials, namely, ∑ D∈Dn ∏ i≥1 t αi(D) i = n ∑ i=1 1 (n− i+ 1)!n,i ( 1!t1, 2!t2, · ...

We consider posets of lattice paths (endowed with a natural order) and begin the study of such structures. We give an algebraic condition to recognize which ones of these posets are lattices. Next we study the class of Dyck lattices (i.e., lattices of Dyck paths) and give a recursive construction for them. The last section is devoted to the presentation of a couple of open problems.

By a Taylor expansion of a generating function, we mean that the remainder of the expansion is a functional of the generating function itself. In this paper, we consider the Taylor expansion for the generating function Bm(t) of them-Catalan numbers. In order to give combinatorial interpretations of the coefficients of these expansions, we study a new collection of partial Grand Dyck paths, that...