scholarly journals Canonical Characters on Quasi-Symmetric Functions and Bivariate Catalan Numbers

10.37236/1872 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
Marcelo Aguiar ◽  
Samuel K. Hsiao
Keyword(s):  
Hopf Algebra ◽  
Symmetric Functions ◽  
Beta Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Explicit Formulas

Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character (Aguiar, Bergeron, and Sottile, math.CO/0310016). We obtain explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasi-symmetric functions. They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers: $$C(m,n)={(2m)!(2n)!\over m!(m+n)!n!}\,.$$ Properties of characters and of quasi-symmetric functions are then used to derive several interesting identities among bivariate Catalan numbers and in particular among Catalan numbers and central binomial coefficients.

scholarly journals Antipode Formulas for some Combinatorial Hopf Algebras

10.37236/5949 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Rebecca Patrias
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Explicit Formulas ◽  
Quasisymmetric Functions ◽  
Noncommutative Symmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Combinatorial Formulas ◽  
K Theory

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.

scholarly journals Computation of several Hessenberg determinants

2020 ◽  
Vol 70 (6) ◽  
pp. 1521-1537
Author(s):  
Feng Qi ◽  
Omran Kouba ◽  
Issam Kaddoura
Keyword(s):  
Linear Algebra ◽  
Simple Formula ◽  
Binomial Coefficients ◽  
Explicit Formulas ◽  
Methods And Techniques

AbstractIn the paper, employing methods and techniques in analysis and linear algebra, the authors find a simple formula for computing an interesting Hessenberg determinant whose elements are products of binomial coefficients and falling factorials, derive explicit formulas for computing some special Hessenberg and tridiagonal determinants, and alternatively and simply recover some known results.

scholarly journals ON DIVISIBILITY OF BINOMIAL COEFFICIENTS

2012 ◽  
Vol 93 (1-2) ◽  
pp. 189-201 ◽  
Author(s):  
ZHI-WEI SUN
Keyword(s):  
Higher Order ◽  
Catalan Numbers ◽  
Binomial Coefficients

AbstractIn this paper, motivated by Catalan numbers and higher-order Catalan numbers, we study factors of products of at most two binomial coefficients.

scholarly journals Colored Trees and Noncommutative Symmetric Functions

10.37236/468 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Matt Szczesny
Keyword(s):  
Hopf Algebra ◽  
Symmetric Functions ◽  
Hall Algebra ◽  
Quasisymmetric Functions ◽  
Noncommutative Symmetric Functions

Let ${\cal CRF}_S$ denote the category of $S$-colored rooted forests, and H$_{{\cal CRF}_S}$ denote its Ringel-Hall algebra as introduced by Kremnizer and Szczesny. We construct a homomorphism from a $K^+_0({\cal CRF}_S)$–graded version of the Hopf algebra of noncommutative symmetric functions to H$_{{\cal CRF}_S}$. Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a $K^+_0({\cal CRF}_S)$–graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals Analogues of the binomial coefficient theorems of Gauss and Jacobi

2016 ◽  
Vol 12 (08) ◽  
pp. 2125-2145
Author(s):  
Abdullah Al-Shaghay ◽  
Karl Dilcher
Keyword(s):  
Gamma Function ◽  
Simple Form ◽  
Binomial Coefficient ◽  
Catalan Numbers ◽  
Binomial Coefficients

The theorems of Gauss and Jacobi that give modulo [Formula: see text] evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo [Formula: see text]. In this paper, we further extend these results to congruences modulo [Formula: see text]. In the process, we prove congruences to arbitrarily high powers of [Formula: see text] for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's [Formula: see text]-adic gamma function. These congruences are of a simple form and involve Catalan numbers as coefficients.

Sign in / Sign up

Export Citation Format

Share Document