Related papers: Generalized Catalan numbers from hypergraphs
Writing the values of Krawtchouk polynomials as matrices, we consider weighted partial sums along columns. For the general case, we find an identity that, in the symmetric case yields a formula for such partial sums. Complete sums of…
We introduce the three-Catalan triangle, highlighting the three-Catalan numbers along with their recurrence relation and combinatorial interpretation, which allows us to establish their log-convexity. Additionally, we prove that the rows of…
We initiate a study of the representation of the symmetric group on the multilinear component of an $n$-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric…
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…
A product difference equation is proved and used for derivation by elementary methods of four combinatorial identities, eight combinatorial identities involving generalized harmonic numbers and eight combinatorial identities involving…
The ECO method and the theory of Catalan-like numbers introduced by Aigner seems two completely unrelated combinatorial settings. In this work we try to establish a bridge between them, aiming at starting a (hopefully) fruitful study on…
Let $C_n$ be the $n$th Catalan number. For any prime $p \geq 5$ we show that the set $\{C_n : n \in \mathbb{N} \}$ contains all residues mod $p$. In addition all residues are attained infinitely often. Any positive integer can be expressed…
The number of flats of a hyperplane arrangement is considered as a generalization of the Bell number and the Stirling number of the second kind. Robert Gill gave the exponential generating function of the number of flats of the extended…
This paper provides the connection between the Hankel transform and aerating transforms of a given integer sequence. Results obtained are used to establish a completely different Hankel transform evaluation of the series reversion of a…
We answer the question in the title in the negative by providing four proofs.
We study the Gaussian Unitary Ensemble (GUE) using noncommutative geometry and the homological framework of the Batalin-Vilkovisky (BV) formalism. Coefficients of the correlation functions in the GUE with respect to the rank $N$ are…
The Concepts of poly-Bernoulli numbers $B_n^{(k)}$, poly-Bernoulli polynomials $B_n^{k}{(t)}$ and the generalized poly-bernoulli numbers $B_{n}^{(k)}(a,b)$ are generalized to $B_{n}^{(k)}(t,a,b,c)$ which is called the generalized…
In this note, we study two generalizations of the Catalan numbers, namely the $s$-Catalan numbers and the spin $s$-Catalan numbers. These numbers first appeared in relation to quantum physics problems about spin multiplicities. We give a…
We show that the Schubert polynomial S_w specializes to the Catalan number C_n when $w=1(n+1)...2$. Several proofs of this result as well as a q-analog are given. An application to the singularities of Schubert varieties is given.
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 normalized volume of the Chan-Robbins-Yuen polytope ($CRY_n$) is the product of consecutive Catalan numbers. The polytope $CRY_n$ has captivated combinatorial audiences for over a decade, as there is no combinatorial proof for its…
In the work [4] tree-rooted planar cubic maps with marked directed edge (not in this tree) were enumerated. The number of such objects with $2n$ vertices is $C_{2n}\cdot C_{n+1}$, where $C_k$ is Catalan number. In this work a marked…
We count the number of countable homogeneous colored linear orderings in $k$ colors. Relatedly, we count the number of countable $C_{n,m}$-homogeneous linear orderings. $C_{n,m}$-homogeneity is a strong homogeneity notion that approximates…
Given a sequence of integers b = (b_0,b_1,b_2,...) one gives a Dyck path P of length 2n the weight wt(P) = b_{h_1} b_{h_2} ... b_{h_n}, where h_i is the height of the ith ascent of P. The corresponding weighted Catalan number is C_n^b =…
This note presents the definition of a proposed generalization of the conchoid at the plane. Known conchoids, such as the Nicomedes and the Lima\c{c}on of Pascal are part of this set. Following the definition, one can generate other…