Related papers: Narayana numbers and Schur-Szego composition

We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating…

We establish a tantalizing symmetry of certain numbers refining the Narayana numbers. In terms of Dyck paths, this symmetry is interpreted in the following way: if $w_{n,k,m}$ is the number of Dyck paths of semilength $n$ with $k$…

We introduce and study a generalization of the Narayana numbers $N_d(n,k) = \frac{1}{n+1} \binom{n+1}{k+1} \binom{ n + (n-k)(d-2)+1}{k}$ for integers $d \geq 2$ and $n,k \geq 0$. This two-parameter array extends the classical Narayana…

The generalized Narayana polynomials $N_{n,m}(x)$ arose from the study of infinite log-concavity of the Boros-Moll polynomials. The real-rootedness of $N_{n,m}(x)$ had been proved by Chen, Yang and Zhang. They also showed that when $n\geq…

The Narayana numbers are $N(n,k) = {1 \over n}{n \choose k}{n \choose {k+1}}$. There are several natural statistics on Dyck paths with a distribution given by N(n,k). We show the equidistribution of Narayana statistics by computing the flag…

In this note, by counting some colored plane trees we obtain several binomial identities. These identities can be viewed as specific evaluations of certain generalizations of the Narayana polynomials. As consequences, it provides…

Let $ (N_n)_{n\ge 0}$ be the Narayana's cow sequence defined by a third-order recurrence relation $ N_0=0,\ N_1= N_2=1 $, and $ N_{n+3}=N_{n+2}+N_n $ for all $ n\ge 0 $. In this paper, we determine all Narayana numbers that are…

We give a simple proof that the the number of Dyck paths of semilength $n$ with $i$ returns to ground level and $j$ peaks is the generalized Narayana number $\frac{i}{n} \binom{n}{j} \binom{n - i - 1}{j - i}$.

Narayana's sequence is a ternary recurrent sequence defined by the recurrence relation $\mathcal{N}_n=\mathcal{N}_{n-1}+\mathcal{N}_{n-3}$ with initial terms $\mathcal{N}_0=0$ and $\mathcal{N}_1=\mathcal{N}_2=\mathcal{N}_3=1$. Let…

The Lucas polynomials, $\{n\}$, are polynomials in $s$ and $t$ given by $\{ n \} = s \{ n-1 \} + t \{ n-2 \}$ for $n \geq 2$ with $ \{ 0 \} = 0$ and $\{ 1 \} = 1$. The lucanomial coefficients, an analogue of the binomial coefficients, are…

We define the Narayana sequence $\{a_n\}_{n\geq 0}$ as the one satisfying the linear recurrence relation $a_n = a_{n-1}+a_{n-3}$ for $n\geq 3$, with initial values $a_0 = 0$ and $a_1 = a_2=1$. In this paper, we fully characterize the…

Every polynomial of the form $P=(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable as Schur-Szeg\H{o} composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, where the numbers $a_i$ are unique up to permutation. We give…

Generalized Euler polynomials ${{\alpha }_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$, where ${{p}_{n}}\left( x \right)$ is the polynomial of degree $n$, are the…

Chen, Deutsch and Elizalde introduced a refinement of the Narayana polynomials by distinguishing between old (leftmost child) and young leaves of plane trees. They also provided a refinement of Coker's formula by constructing a bijection.…

In this note, by employing a nice property of semicircular distributions, we derive some identities for the Narayana polynomial and its derivatives.

The generating functions of some sequences of Catalan numbers and Narayana polynomials have simple expansions as continued fractions of Jacobi type. We give an overview of these facts and prove analogous results for q-Narayana polynomials…

The $n^{\text{th}}$ small Schr\"oder number is $s(n) = \sum_{k \geq 0} s(n,k)$, where $s(n,k)$ denotes the number of plane rooted trees with $n$ leaves and $k$ internal nodes that each has at least two children. In this manuscript, we focus…

Each degree $n+k$ polynomial of the form $(x+1)^k(x^n+c_1x^{n-1}+\cdots +c_n)$, $k\in \mathbb{N}$, is representable as Schur-Szeg\H{o} composition of $n$ polynomials of the form $(x+1)^{n+k-1}(x+a_j)$. We study properties of the affine…

Each degree $n$ polynomial in one variable of the form $(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ is representable in a unique way as a Schur-Szeg\H{o} composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, see \cite{Ko1},…

In this paper, we focus on Narayana numbers which can be written as a products of four repdigits in base $g$, where $g$ is an integer with $g\geq2$. We prove that for $g$ between $2$ and $12$, there are finitely many of these numbers.…