Related papers: Catalan pairs and Fishburn triples

Certain $*$-semigroups are associated with the universal $C^*$-algebra generated by a partial isometry, which is itself the universal $C^*$-algebra of a $*$-semigroup. A fundamental role for a $*$-structure on a semigroup is emphasized, and…

Motivated by Vassiliev's knot invariants, Stoimenow introduced a special class of matchings, now known as Stoimenow matchings. These matchings have since been linked to various combinatorial structures enumerated by the Fishburn numbers. In…

In this paper we show that many well-known counting coefficients, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the central Delannoy numbers are Hausdorff moment sequences in a unified approach. In…

The notion of the Fibonacci cobweb poset from [1] has been naturally extended to any admissible sequence $F$ in [2] where it was also recognized that the celebrated prefab notion of Bender and Goldman [3] - (see also [4,5]) - admits such an…

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…

Building upon a recent formula for $(3,m)$-Catalan polynomials, we describe a formula for $(3,m)$-Hikita polynomials in terms related to Catalan polynomials. This formula shows a surprising relation among coefficients of Hikita polynomials…

Generalized Pascal matrix whose elements are generalized binomial coefficients is included in the group of generalized Riordan arrays. There is a special set of generalized Riordan arrays defined by parameter $q$. If $q=0$, they are…

In this note we investigate the problem of finding pairs of Pythagorean triangles $(a, b, c), (A, B, C)$, with given catheti ratios $A/a, B/b$. In particular, we prove that there are infinitely many essentially different ("non-similar")…

The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuss-Catalan generalizations. We provide new…

We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for…

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…

We present a $q$-analog of the super Catalan number $(2m)!(2n)!/2m!n!(m+n)!$, which also generalizes the $q$-Catalan numbers $c_n(\lambda)$, due to F\"urlinger and Hofbauer, for $\lambda=0$ and $\lambda=1$. We give a combinatorial…

Linearly distributive categories were introduced to model the tensor/par fragment of linear logic, without resorting to the use of negation. Linear bicategories are the bicategorical version of linearly distributive categories. Essentially,…

In this paper, we exploit the combinatorics and geometry of triangulations of products of simplices to derive new results in the context of Catalan combinatorics of $\nu$-Tamari lattices. In our framework, the main role of "Catalan objects"…

We use quasi-orders to describe the structure of C-groups. We do this by associating a quasi-order to each compatible C-relation of a group, and then give the structure of such quasi-ordered groups. We also reformulate in terms of…

Using generalized binomial coefficient identities and some results of John Dougall, we derive some families of series involving the cubes of Catalan numbers. We also establish a family of series containing fourth powers of Catalan numbers.…

In this note, we provide bijective proofs of some identities involving the Bell number, as previously requested. Our arguments may be extended to yield a generalization in terms of complete Bell polynomials. We also provide a further…

Recently, Jel\'{i}nek derived that the number of self-dual interval orders of reduced size $n$ is twice the number of row-Fishburn matrices of size $n$ by using generating functions. In this paper, we present a bijective proof of this…

Analogical proportions are statements of the form "$a$ is to $b$ as $c$ is to $d$", which expresses that the comparisons of the elements in pair $(a, b)$ and in pair $(c, d)$ yield similar results. Analogical proportions are creative in the…

The super Catalan numbers $T(m,n)=(2m)!(2n)!/2m!n!(m+n)!$ are integers which generalize the Catalan numbers. With the exception of a few values of $m$, no combinatorial interpretation in known for $T(m,n)$. We give a weighted interpretation…