scholarly journals The translated Whitney–Lah numbers: generalizations and q-analogues

2020 ◽  
Vol 26 (4) ◽  
pp. 80-92
Author(s):  
Mahid M. Mangontarum ◽  
Keyword(s):  
Whitney Numbers ◽  
Combinatorial Formulas

In this paper, we derive some combinatorial formulas for the translated Whitney–Lah numbers which are found to be generalizations of already-existing identities of the classical Lah numbers, including the well-known Qi’s formula. Moreover, we obtain q-analogues of the said formulas and identities by establishing similar properties for the translated q-Whitney numbers.

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

Hankel Transform of the Type 2 (p,q)-Analogue of r-Dowling Numbers

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 343
Author(s):  
Roberto Corcino ◽  
Mary Ann Ritzell Vega ◽  
Amerah Dibagulun
Keyword(s):  
Hankel Transform ◽  
Convolution Type ◽  
Combinatorial Interpretation ◽  
Whitney Numbers

In this paper, type 2 (p,q)-analogues of the r-Whitney numbers of the second kind is defined and a combinatorial interpretation in the context of the A-tableaux is given. Moreover, some convolution-type identities, which are useful in deriving the Hankel transform of the type 2 (p,q)-analogue of the r-Whitney numbers of the second kind are obtained. Finally, the Hankel transform of the type 2 (p,q)-analogue of the r-Dowling numbers are established.

Whitney Numbers

1987 ◽  
pp. 139-160 ◽  
Author(s):  
Martin Aigner
Keyword(s):  
Whitney Numbers

scholarly journals New Combinatorial Formulas for Cluster Monomials of Type $A$ Quivers

10.37236/6464 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li ◽  
Ba Nguyen
Keyword(s):  
Cluster Algebra ◽  
Theta Functions ◽  
Type A ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Combinatorial Formula ◽  
Natural Basis ◽  
Combinatorial Formulas ◽  
New Formula

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type $A$ cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the $T$-paths and of the perfect matchings in a snake diagram. For cluster variables of a type $A$ cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.

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 The $Z$-Polynomial of a Matroid

10.37236/7105 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Nicholas Proudfoot ◽  
Yuan Xu ◽  
Ben Young
Keyword(s):  
Closed Formula ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Whitney Numbers ◽  
Alternating Sums ◽  
Cohomological Interpretation

We introduce the $Z$-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the $Z$-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion, obtaining a closed formula for Kazhdan-Lusztig coefficients as alternating sums of multi-indexed Whitney numbers. For realizable matroids, we give a cohomological interpretation of the $Z$-polynomial in which the symmetry is a manifestation of Poincaré duality.

Some polynomials associated with the $${\varvec{r}}$$ r -Whitney numbers

2018 ◽  
Vol 128 (3) ◽  
Author(s):  
Cristina B Corcino ◽  
Roberto B Corcino ◽  
István Mező ◽  
José L Ramírez
Keyword(s):  
Whitney Numbers
Sign in / Sign up

Export Citation Format

Share Document