Related papers: A Note on Generalized Narayana Numbers

Mizuno provided 19 examples of generalized rank three Nahm sums with symmetrizer $\mathrm{diag}(1,1,2)$ which are conjecturally modular. We confirm their modularity by establishing Rogers--Ramanujan type identities of index $(1,1,2)$ for…

Let $f$ be an arithmetical function. The matrix $[f(i,j)]_{n\times n}$ given by the value of $f$ in greatest common divisor of $(i,j)$, $f\big((i,j)\big)$ as its $i,\; j$ entry is called the greatest common divisor (GCD) matrix. We consider…

Pappe, Paul, and Schilling introduced two combinatorial statistics, depth and ddinv, associated with classical Dyck paths, and proved that the distributions of (area, depth) and (dinv, ddinv) are $q,t$-symmetric by constructing an…

Let $c\in (0, 1]$ be a real number and let $n$ be a sufficiently large integer. We prove that every $n$-vertex $c n$-regular graph $G$ contains a collection of $\lfloor 1/c \rfloor$ paths whose union covers all but at most $o(n)$ vertices…

We consider a bi-criteria generalization of the pathwidth problem, where, for given integers $k,l$ and a graph $G$, we ask whether there exists a path decomposition $\cP$ of $G$ such that the width of $\cP$ is at most $k$ and the number of…

We introduce square diagrams that represent numerical semigroups and we obtain an injection from the set of numerical semigroups into the set of Dyck paths.

We present a new bijection between variants of $m$-Dyck paths (paths with steps in $\{+1,-m\}$ starting and ending at height $0$ and remaining at non-negative height), which generalizes a classical bijection between Dyck prefixes and…

In this paper, we investigate the properties of recurrent planar Markov random walks. More precisely, we study the set of recurrent points with the use of local limit theorems. The Nagaev-Guivarc'h spectral method provides several examples…

Let N_d be the number of degree d, nodal, rational plane curves through 3d-1 points in the complex projective plane. The number of degree d>=3, nodal, elliptic plane curves with a fixed (general) j-invariant through 3d-1 points is found to…

Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…

The notion of symmetric and asymmetric peaks in Dyck paths was introduced by Fl\'orez and Rodr\'{\i}guez, who counted the total number of such peaks over all Dyck paths of a given length. In this paper we generalize their results by giving…

It is a longstanding open problem to find a bijection exchanging area and bounce statistics on Dyck paths. We settle this problem for an exponentially large subset of Dyck paths via an explicit bijection. Moreover, we prove that this…

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 paper we introduce I^K-convergence which is a common generalization of the I^K-convergence of sequences, double sequences and nets. We show that many results that were shown before for these special cases are true for the…

Let $A$ be an algebra and let $X$ be an $A$-bimodule. A $\Bbb C-$linear mapping $d:A \to X$ is called a generalized Jordan derivation if there exists a Jordan derivation (in the usual sense) $\delta:A \to X$ such that…

In this paper, motivated by applications in computer graphics and animation, we study the numerical methods for checking $C^k-$regularity of vector multivariate subdivision schemes with dilation 2I. These numerical methods arise from the…

Let $\{N_m\}_{m\ge0}$ be the Narayana's cows sequence given by $N_0=0$, $N_1=1=N_2=1$ and \[ N_{m+3}=N_{m+2}+N_m,\quad \text{ for }\; m\geq 0 \] and let $\{F_n\}_{n\ge0}$ be the Fibonacci sequence. In this paper we solve explicitely the…

Given a graph $G(V, E)$ and a positive integer $k$ ($k \geq 1$), a simple path on $k$ vertices is a sequence of $k$ vertices in which no vertex appears more than once and each consecutive pair of vertices in the sequence are connected by an…

A subset of vertices of a graph $G$ is a general position set if no triple of vertices from the set lie on a common shortest path in $G$. In this paper we introduce the general position polynomial as $\sum_{i \geq 0} a_i x^i$, where $a_i$…

In 1983 Kalai proved an incredible generalisation of Cayley's formula for the number of trees on a labelled vertex set to a formula for a class of $r$-dimensional simplicial complexes. These simplicial complexes generalise trees by means of…