Related papers: Catalan number sequences and generalized action gr…
We introduce an inductively defined sequence of directed graphs and prove that the number of edges added at step $k$ is equal to the $k$th Catalan number. Furthermore, we establish an isomorphism between the set of edges adjoined at step…
The Catalan numbers $C_n$ are an extremely well-studied sequence of numbers that appear as the answer to many combinatorial problems. Two generalizations of these numbers that have been studied are the Fuss-Catalan numbers and the…
We show that families of action graphs, with initial graphs which are linear of varying length, give rise to self-convolutions of the Catalan sequence. We prove this result via a comparison with planar rooted forests with a fixed number of…
This paper explores the properties of directed graphs, termed generalized action graphs, which exhibit a strong connection to certain number sequences. Focusing on the structural and combinatorial aspects, we investigate the conditions…
The Catalan numbers constitute one of the most important sequences in combinatorics. Catalan objects have been generalized in various directions, including the classical Fuss-Catalan objects and the rational Catalan generalization of…
The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.
We first observe that the relations of the canonical generating isometries of the Cuntz algebra ${\cal O}_N$ are naturally related to the $N$-colored Catalan numbers. For a directed graph $G$, we generalize the Catalan numbers by using the…
The Catalan numbers (C_n)_{n >= 0} = 1,1,2,5,14,42,... form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting…
We consider Catalan-pair graphs, a family of graphs that can be viewed as representing certain interactions between pairs of objects which are enumerated by the Catalan numbers. In this paper we study random Catalan-pair graphs and deduce…
In the paper, the authors analytically generalize the Catalan numbers in combinatorial number theory, establish an integral representation of the analytic generalization of the Catalan numbers by virtue of Cauchy's integral formula in the…
The genus--g Fuss--Catalan (FC) number counts the number of ways to obtain a genus-g surface by identifying the edges of a pn--gon via p-valent hyperedges. For p=2 these are the genus--g Catalan numbers which are generated as the trace…
Path pairs are a modification of parallelogram polyominoes that provide yet another combinatorial interpretation of the Catalan numbers. More generally, the number of path pairs of length $n$ and distance $\delta$ corresponds to the…
We present an exposition on the Fuss--Catalan numbers, which are a generalization of the well known Catalan numbers. The literature on the subject is scattered (especially for the case of multiple independent parameters, as will be…
Catalan numbers $C(n)=\frac{1}{n+1}{2n\choose n}$ enumerate binary trees and Dyck paths. The distribution of paths with respect to their number $k$ of factors is given by ballot numbers $B(n,k)=\frac{n-k}{n+k}{n+k\choose n}$. These integers…
We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic:…
Recent work of the first author, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov--Rozansky knot homology produces a family of polynomials in $q$ and $t$ labeled by integer sequences. These polynomials can be…
We show that recent determinant evaluations involving Catalan numbers and generalisations thereof have most convenient explanations by combining the Lindstr\"om-Gessel-Viennot theorem on non-intersecting lattice paths with a simple…
We give an example of two non isomorphic coordinate rings of a special kind of convex polyominoes whose Cohen-Macaulay types are generalized Fuss-Catalan numbers. We further provide a determinantal formula for these numbers.
We refine Catalan numbers and Fu{\ss}-Catalan numbers by introducing colour statistics for triangulations of polygons and $d$-dimensional generalisations there-of which we call Fu{\ss}-Catalan complexes. Our refinements consist in showing…
It is well known that permutations avoiding any 3-length pattern are enumerated by the Catalan numbers. If the three patterns 123, 132 and 213 are avoided at the same time we obtain a class of permutations enumerated by the Fibonacci…