A Combinatorial Interpretation of the Seidel Generation of Genocchi Numbers

1980 ◽  
pp. 77-87 ◽  
Author(s):  
Duminique Dumont ◽  
Gérard Viennot
Keyword(s):  
Combinatorial Interpretation ◽  
Genocchi Numbers

scholarly journals Combinatorial Study of Dellac Configurations and $q$-Extended Normalized Median Genocchi Numbers

10.37236/4068 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Ange Bigeni
Keyword(s):  
Flag Varieties ◽  
Combinatorial Interpretation ◽  
Genocchi Numbers ◽  
Degenerate Flag Varieties

In two recent papers, Feigin proved that the Poincaré polynomials of the degenerate flag varieties have a combinatorial interpretation through Dellac configurations, and related them to the $q$-extended normalized median Genocchi numbers $\bar{c}_n(q)$ introduced by Han and Zeng, mainly by geometric considerations. In this paper, we give combinatorial proofs of these results by constructing statistic-preserving bijections between Dellac configurations and two other combinatorial models of $\bar{c}_n(q)$.

scholarly journals Combinatorial Interpretations of the Kreweras Triangle in Terms of Subset Tuples

10.37236/7531 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Ange Bigeni
Keyword(s):  
Combinatorial Interpretation ◽  
Genocchi Numbers ◽  
Combinatorial Interpretations ◽  
Dumont Permutations

We show how the combinatorial interpretation of the normalized median Genocchi numbers in terms of multiset tuples, defined by Hetyei in his study of the alternation acyclic tournaments, is bijectively equivalent to previous models like the normalized Dumont permutations or the Dellac configurations, and we extend the interpretation to the Kreweras triangle.

scholarly journals Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

2021 ◽  
Vol 111 (3) ◽  
Author(s):  
Massimo Gisonni ◽  
Tamara Grava ◽  
Giulio Ruzza
Keyword(s):  
Generating Functions ◽  
Combinatorial Interpretation ◽  
Hurwitz Numbers ◽  
Jacobi Ensemble ◽  
Matrix Ensembles ◽  
Wilson Polynomials ◽  
Unitary Ensemble ◽  
Topological Expansion

AbstractWe express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulæ for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

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.

scholarly journals Some New Identities on the -Genocchi Numbers and Polynomials with Weight

2011 ◽  
Vol 2011 ◽  
pp. 1-6
Author(s):  
Seog-Hoon Rim ◽  
Joohee Jeong
Keyword(s):  
Genocchi Numbers ◽  
New Type

We construct a new type of -Genocchi numbers and polynomials with weight . From these -Genocchi numbers and polynomials with weight , we establish some interesting identities and relations.

scholarly journals Combinatorics of k-shapes and Genocchi numbers

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Florent Hivert ◽  
Olivier Mallet
Keyword(s):  
Symmetric Functions ◽  
New Model ◽  
Work In Progress ◽  
Genocchi Numbers ◽  
Combinatorial Model ◽  
International Audience

International audience In this paper we present a work in progress on a conjectural new combinatorial model for the Genocchi numbers. This new model called irreducible k-shapes has a strong algebraic background in the theory of symmetric functions and leads to seemingly new features on the theory of Genocchi numbers. In particular, the natural q-analogue coming from the degree of symmetric functions seems to be unknown so far. Dans cet article, nous présentons un travail en cours sur un nouveau modèle combinatoire conjectural pour les nombres de Genocchi. Ce nouveau modèle est celui des k-formes irréductibles, qui repose sur de solides bases algébriques en lien avec la théorie des fonctions symétriques et qui conduit à des aspects apparemment nouveaux de la théorie des nombres de Genocchi. En particulier, le q-analogue naturel venant du degré des fonctions symétriques semble inconnu jusqu'ici.

scholarly journals The Cube Recurrence

10.37236/1826 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Gabriel D. Carroll ◽  
David Speyer
Keyword(s):  
Recurrence Relation ◽  
Planar Graphs ◽  
Perfect Matchings ◽  
Laurent Polynomials ◽  
Combinatorial Interpretation ◽  
Combinatorial Model

We construct a combinatorial model that is described by the cube recurrence, a quadratic recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in ${\Bbb Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky about the structure of these polynomials, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences, including the Somos-6 and Somos-7 sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs.

scholarly journals Multivariate Eulerian Polynomials and Exclusion Processes

2016 ◽  
Vol 25 (4) ◽  
pp. 486-499 ◽  
Author(s):  
P. BRÄNDÉN ◽  
M. LEANDER ◽  
M. VISONTAI
Keyword(s):  
Finite Number ◽  
Exclusion Process ◽  
Partition Functions ◽  
Asymmetric Exclusion Process ◽  
Negative Dependence ◽  
Combinatorial Interpretation ◽  
Exclusion Processes ◽  
Eulerian Polynomials ◽  
The Third ◽  
Dependence Properties

We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and coloured permutations. The corresponding expressions of the multivariate partition functions are then related to multivariate generalisations of Eulerian polynomials for coloured permutations considered recently by N. Williams and the third author, and others. We also discuss stability and negative dependence properties satisfied by the partition functions.

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