Related papers: Catalan structures and Catalan pairs
We give a combinatorial interpretation using lattice paths for the super Catalan number $S(m, m+s)$ for $s \leq 3$ and a separate interpretation for $s = 4$.
The cartesian structure possessed by relations, spans, profunctors, and other such morphisms is elegantly expressed by universal properties in double categories. Though cartesian double categories were inspired in part by the older program…
A Catalan magma is a unique factorisation normed magma with only one irreducible element. The \val partitions the base set into subsets enumerated by Catalan numbers. The primary theorem characterises the conditions which a set a with…
Let $G$ be a finite, simple, connected graph. An arithmetical structure on $G$ is a pair of positive integer vectors $\mathbf{d},\mathbf{r}$ such that $(\mathrm{diag}(\mathbf{d})-A)\mathbf{r}=0$, where $A$ is the adjacency matrix of $G$. We…
We introduce the $q,t$-Catalan measures, a sequence of piece-wise polynomial measures on $\mathbb{R}^2$. These measures are defined in terms of suitable area, dinv, and bounce statistics on continuous families of paths in the plane, and…
We answer a question of Simental by providing a combinatorial interpretation of a formula which generalizes rational Catalan numbers and which appears in the study of Springer fibers. We provide an interpretation in terms of binary…
We demonstrate that a natural construction based on the two notions of insertion of a strand and finding the preimages of Temperley-Lieb algebra generators give an inductive means to generate all link patterns of a given number of strands.…
We use Riordan array theory to give characterizations of the Borel triangle and its associated polynomial sequence. We show that the Borel polynomials are the moment sequence for a family of orthogonal polynomials whose coefficient array is…
Kantor pairs arise naturally in the study of 5-graded Lie algebras. In this article, we begin the study of simple Kantor pairs of arbitrary dimension. We introduce Weyl images of Kantor pairs and use them to construct examples of Kantor…
Let \(A=(A,\star)\) be a finite binary algebra, not necessarily associative. For each \(n\geq 1\), every full binary bracketing on \(x_1,\dots,x_n\) determines an \(n\)-ary term operation on \(A\), and hence an evaluation word obtained by…
The sets used to construct other mathematical objects are pure sets, which means that all of their elements are sets, which are themselves pure. One set may therefore be within another, not as an element, but as an element of an element, or…
The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$.…
We indicate a way of distinguishing between structures, for which, we call two structures distinguishable. Roughly, being distinguishable means that they differ in the number of realizations each gives for some formula. Being…
In this note, we derive an alternative recursive formula for the sums of powers of integers involving the Stirling numbers of the first kind. As a remarkable by-product, we provide a non-recursive definition of the Catalan numbers.
Guo-Niu Han [arXiv:2006.14070 [math.CO]] has introduced a new combinatorial object named standard puzzle. We use digraphs to show the relations between numbers in standard puzzles and propose a skeleton model. By this model, we solve the…
This paper is concerned with the foundations of the Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions by inductive data types. CAC generalizes inductive types equipped with higher-order primitive…
It is well-known fact that there exists 1-1 correspondence between so-called double (or flou) sets and intuitionistic sets (also known as orthopairs). At first glance, these two concepts seem to be irreconcilable. However, one must remember…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
Given a combinatorial structure, a ``twin'' is a pair of disjoint substructures which are isomorphic (or look the same in some sense). In recent years, there have been many problems about finding large twins in various combinatorial…
Given two quasi-definite moment functionals, the corresponding orthogonal polynomial systems satisfy an algebraic differential relation(called an extended coherent pair). We study generalizing extended coherent pairs that unify extended…