Related papers: Catalan pairs and Fishburn triples
In type A, the q,t-Fuss -Catalan numbers can be defined as a bigraded Hilbert series of a module associated to the symmetric group S_n. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured…
A finitization of the Catalan numbers $ C_n $ can be defined as Euler characteristics of an algebraic structure. We conjecture the existence of a $ q $-deformed version of such structure, and provide evidence for the first two non-trivial…
The focus of this paper is the study of generalized Fibonacci polynomials and Fibonomial coefficients. The former are polynomials {n} in variables s and t given by {0} = 0, {1} = 1, and {n} = s{n-1}+t{n-2} for n ge 2. The latter are defined…
The Fuss-Catalan numbers are a generalization of the Catalan numbers. They enumerate a large class of objects and in particular m-Dyck paths and m+1-ary trees. Recently, F. Bergeron defined an analogue for generic m of the Tamari order on…
We introduce (dual) strongly relative CS-Rickart objects in abelian categories, as common generalizations of (dual) strongly relative Rickart objects and strongly extending (lifting) objects. We give general properties, and we study direct…
The Fishburn numbers, $\xi(n),$ are defined by a formal power series expansion $$ \sum_{n=0}^\infty \xi(n)q^n = 1 + \sum_{n=1}^\infty \prod_{j=1}^n (1-(1-q)^j). $$ For half of the primes $p$, there is a non--empty set of numbers $T(p)$…
Correlation functions in a dynamic quartic matrix model are obtained from the two-point function through a recurrence relation. This paper gives the explicit solution of the recurrence by mapping it bijectively to a two-fold nested…
The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of $\mathcal{F}$-quotients in a…
We use the inversion of coefficient arrays to define dual polynomials to the Fibonacci and Catalan-Fibonacci polynomials, and we explore the properties of these new polynomials sequences. Many of the arrays involved are Riordan arrays.…
We consider compositions of natural numbers when there are different types of each natural number. Several recursions as well as some closed formulas for the number of compositions is derived. We also find its relationships with some known…
The main goal of this paper is to assign two combinatorial interpretations to the elements of the Rascal Triangle defined by Angorro et al. The first interpretation involves counting ascents in binary words, while the second interpretation…
We introduce the notion of a Mahonian pair. Consider the set, P^*, of all words having the positive integers as alphabet. Given finite subsets S,T of P^*, we say that (S,T) is a Mahonian pair if the distribution of the major index, maj,…
In this paper, we extend the recently introduced concept of partially dual ribbon graphs to graphs. We then go on to characterize partial duality of graphs in terms of bijections between edge sets of corresponding graphs. This result…
Ascent sequences form a central class of combinatorial objects, as they are in bijection with several important families such as (2+2)-free posets, Stoimenow matchings, and other Fishburn objects, and are enumerated by the Fishburn numbers.…
In 1996, Stanley extended the classical Catalan arrangement and semiorder arrangement, which are called the Catalan-type arrangement $\mathcal{C}_{n,A}$ and the semiorder-type arrangement $\mathcal{C}_{n,A}^*$ in this paper. By establishing…
We define duality triples and duality pairs in compactly generated triangulated categories and investigate their properties. This enables us to give an elementary way to determine whether a class is closed under pure subobjects, pure…
In this paper, bicomplex k-Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex k-Fibonacci quaternions which are connected with bicomplex numbers and k-Fibonacci numbers are investigated. Furthermore, the…
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to mean that some external…
A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group $G$. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over…