On some properties of generalized Narayana numbers

Abstract For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

Download Full-text

On the new Narayana polynomials, the Gauss Narayana numbers and their polynomials

Asian-European Journal of Mathematics ◽

10.1142/s179355712150100x ◽

2020 ◽

pp. 2150100

Author(s):

Engi̇n Özkan ◽

Bahar Kuloğlu

Keyword(s):

Hankel Transform ◽

Pell Numbers ◽

Pascal’S Triangle ◽

Narayana Numbers ◽

Definition Of ◽

Pascal's Triangle ◽

The Relationship ◽

Derivatives Of

We give a new definition of Narayana polynomials and show that there is a relationship between the coefficient of the new Narayana polynomials and Pascal’s triangle. We define the Gauss Narayana numbers and their polynomials. Then we show that there is a relationship between the Gauss Narayana polynomials and the new Narayana polynomials. Also, we show that there is a relationship between the derivatives of the new Narayana polynomials and Pascal’s triangle. We also explain the relationship between the new Narayana polynomials and the known Pell numbers. Finally, we give the Hankel transform of the new Narayana polynomials.

Download Full-text

Narayana numbers and Schur–Szegő composition

Journal of Approximation Theory ◽

10.1016/j.jat.2008.10.013 ◽

2009 ◽

Vol 161 (2) ◽

pp. 464-476 ◽

Cited By ~ 6

Author(s):

Vladimir P. Kostov ◽

Andrei Martínez-Finkelshtein ◽

Boris Z. Shapiro

Keyword(s):

Narayana Numbers

Download Full-text

Distance Fibonacci Polynomials by Graph Methods

Symmetry ◽

10.3390/sym13112075 ◽

2021 ◽

Vol 13 (11) ◽

pp. 2075

Author(s):

Dominik Strzałka ◽

Sławomir Wolski ◽

Andrzej Włoch

Keyword(s):

Initial Conditions ◽

Symmetric Matrix ◽

Fibonacci Polynomials ◽

Pascal’S Triangle ◽

Graph Interpretation ◽

Symmetric Structure ◽

Binomial Formula ◽

Narayana Numbers ◽

Pascal's Triangle

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions.

Download Full-text

Two Statistics Linking Dyck Paths and Non-crossing Partitions

The Electronic Journal of Combinatorics ◽

10.37236/570 ◽

2011 ◽

Vol 18 (1) ◽

Author(s):

Haijian Zhao ◽

Zheyuan Zhong

Keyword(s):

Catalan Numbers ◽

Dyck Paths ◽

Narayana Numbers

We introduce a pair of statistics, maj and sh, on Dyck paths and show that they are equidistributed. Then we prove that this maj is equivalent to the statistics $ls$ and $rb$ on non-crossing partitions. Based on non-crossing partitions, we give the most obvious $q$-analogue of the Narayana numbers and the Catalan numbers.

Download Full-text

Identities Derived from Noncrossing Partitions of Type $B$

The Electronic Journal of Combinatorics ◽

10.37236/616 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 3

Author(s):

William Y. C. Chen ◽

Andrew Y. Z. Wang ◽

Alina F. Y. Zhao

Keyword(s):

Type B ◽

Noncrossing Partitions ◽

Narayana Numbers ◽

Combinatorial Constructions

Based on weighted noncrossing partitions of type $B$, we obtain type $B$ analogues of Coker's identities on the Narayana polynomials. A parity reversing involution is given for the alternating sum of Narayana numbers of type $B$. Moreover, we find type $B$ analogues of the refinements of Coker's identities due to Chen, Deutsch and Elizalde. By combinatorial constructions, we provide type $B$ analogues of three identities of Mansour and Sun also on the Narayana polynomials.

Download Full-text

A New Bijection Between RNA Secondary Structures and Plane Trees and its Consequences

The Electronic Journal of Combinatorics ◽

10.37236/8540 ◽

2019 ◽

Vol 26 (4) ◽

Author(s):

Ricky Xiaofeng Chen

Keyword(s):

Biological Significance ◽

Secondary Structures ◽

Path Decomposition ◽

Rna Secondary Structures ◽

Plane Trees ◽

Young Leaves ◽

Narayana Numbers ◽

Horizontal Fiber

In this paper, we first present a new bijection between RNA secondary structures and plane trees. Combined with the Schmitt-Waterman bijection between these objects, we then obtain a bijection on plane trees that relates the horizontal fiber decomposition associated to internal vertices to the degrees of odd-level vertices while the vertical path decomposition associated to leaves is related to the degrees of even-level vertices. To the best of our knowledge, only the former relation (i.e., horizontal vs odd-level) due to Deutsch is known. As a consequence, we obtain enumeration results for various classes of plane trees, e.g., refining the Narayana numbers and the enumeration involving young leaves due to Chen, Deutsch and Elizalde, and counting a newly introduced `vertical' version of $k$-ary trees. The enumeration results can be also formulated in terms of RNA secondary structures with certain parameterized features, which might have some biological significance.

Download Full-text