scholarly journals The ${1/k}$-Eulerian Polynomials

10.37236/16 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Carla D. Savage ◽  
Gopal Viswanathan
Keyword(s):  
Geometric Realization ◽  
Continuous Variable ◽  
Lecture Hall ◽  
Combinatorial Interpretation ◽  
Bivariate Polynomials ◽  
Eulerian Numbers ◽  
Eulerian Polynomials ◽  
Defining Relations ◽  
Generalized Inversion ◽  
Lecture Hall Partitions

We use the theory of lecture hall partitions to define a generalization of the Eulerian polynomials, for each positive integer $k$.  We show that these ${1}/{k}$-Eulerian polynomials have a simple combinatorial interpretation in terms of a single statistic on generalized inversion sequences. The theory provides a geometric realization of the polynomials as the $h^*$-polynomials of $k$-lecture hall polytopes. Many of the defining relations of the Eulerian polynomials have natural ${1}/{k}$-generalizations.  In fact,  these properties extend to a bivariate generalization obtained by replacing  ${1}/{k}$ by a  continuous variable. The bivariate polynomials have appeared in the work of Carlitz, Dillon, and Roselle on Eulerian numbers of higher order and, more recently, in the theory of rook polynomials.

scholarly journals Lecture Hall Partitions and the Affine Hyperoctahedral Group

10.37236/7201 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Christopher R. H. Hanusa ◽  
Carla D. Savage
Keyword(s):  
Generating Function ◽  
The Other ◽  
Lecture Hall ◽  
Hyperoctahedral Group ◽  
Lecture Hall Partitions ◽  
New View

In 1997 Bousquet-Mélou and Eriksson introduced lecture hall partitions as the inversion vectors of elements of the parabolic quotient $\widetilde{C}/C$.  We provide a new view of their correspondence that allows results in one domain to be translated into the other.  We determine the equivalence between combinatorial statistics in each domain and use this correspondence to translate certain generating function formulas on lecture hall partitions to new observations about $\widetilde{C}/C$.

scholarly journals New Eulerian Numbers of Type $D$

10.37236/5514 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Anna Borowiec ◽  
Wojciech Młotkowski
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Probability Distributions ◽  
Type A ◽  
Eulerian Numbers ◽  
Eulerian Polynomials ◽  
Type D

We introduce a new array of type $D$ Eulerian numbers, different from that studied by Brenti, Chow and Hyatt. We find in particular the recurrence relation, Worpitzky formula and the generating function. We also find the probability distributions whose moments are Eulerian polynomials of type $A$, $B$ and $D$.

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.

scholarly journals s-Lecture hall partitions, self-reciprocal polynomials, and Gorenstein cones

2014 ◽  
Vol 36 (1-2) ◽  
pp. 123-147 ◽  
Author(s):  
Matthias Beck ◽  
Benjamin Braun ◽  
Matthias Köppe ◽  
Carla D. Savage ◽  
Zafeirakis Zafeirakopoulos
Keyword(s):  
Lecture Hall ◽  
Lecture Hall Partitions

scholarly journals General Eulerian Numbers and Eulerian Polynomials

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Tingyao Xiong ◽  
Hung-Ping Tsao ◽  
Jonathan I. Hall
Keyword(s):  
Arithmetic Progressions ◽  
Eulerian Numbers ◽  
Eulerian Polynomials

We will generalize the definitions of Eulerian numbers and Eulerian polynomials to general arithmetic progressions. Under the new definitions, we have been successful in extending several well-known properties of traditional Eulerian numbers and polynomials to the general Eulerian polynomials and numbers.

scholarly journals Eulerian Numbers Associated with Arithmetical Progressions

10.37236/7182 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
José L. Ramírez ◽  
Sergio N. Villamarin ◽  
Diego Villamizar
Keyword(s):  
Generating Functions ◽  
Combinatorial Identities ◽  
Combinatorial Interpretation ◽  
Eulerian Numbers ◽  
Signed Permutations ◽  
Whitney Numbers

In this paper, we give a combinatorial interpretation of the $r$-Whitney-Eulerian numbers by means of coloured signed permutations. This sequence is a generalization of the well-known Eulerian numbers and it is connected to $r$-Whitney numbers of the second kind. Using generating functions, we provide some combinatorial identities and the log-concavity property. Finally, we show some basic congruences involving the $r$-Whitney-Eulerian numbers.

scholarly journals Mixed volumes of hypersimplices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Mixed Volumes ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Nous Montrons ◽  
Set Of Permutations ◽  
International Audience

International audience In this extended abstract we consider mixed volumes of combinations of hypersimplices. These numbers, called mixed Eulerian numbers, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type $B$ analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers. Dans ce résumé étendu nous considérons les volumes mixtes de combinaisons d’hyper-simplexes. Ces nombres, appelés les nombres Eulériens mixtes, ont été pour la première fois étudiés par A. Postnikov, et il a été montré qu’ils satisfont à de nombreuses propriétés reliées aux nombres Eulériens, au nombres de Catalan, aux coefficients binomiaux, etc. Nous donnons une interprétation combinatoire générale des nombres Eulériens mixtes, et nous prouvons combinatoirement les propriétés mentionnées ci-dessus. En particulier, nous montrons que chaque nombre Eulérien mixte compte les éléments d’un certain sous-ensemble de l’ensemble des permutations $S_n$. Nous établissons également plusieurs nouvelles propriétés des nombres Eulériens mixtes grâce à notre méthode. Pour finir, nous introduisons une généralisation en type $B$ des nombres Eulériens mixtes, et nous en donnons une interprétation combinatoire analogue.

scholarly journals ON q-SERIES IDENTITIES ARISING FROM LECTURE HALL PARTITIONS

2009 ◽  
Vol 05 (02) ◽  
pp. 327-337 ◽  
Author(s):  
GEORGE E. ANDREWS ◽  
SYLVIE CORTEEL ◽  
CARLA D. SAVAGE
Keyword(s):  
Lecture Hall ◽  
Lecture Hall Partitions

In this paper, we highlight two q-series identities arising from the "five guidelines" approach to enumerating lecture hall partitions and give direct, q-series proofs. This requires two new finite corollaries of a q-analog of Gauss's second theorem. In fact, the method reveals stronger results about lecture hall partitions and anti-lecture hall compositions that are only partially explained combinatorially.

scholarly journals Diagonal Checker-jumping and Eulerian Numbers for Color-signed Permutations

10.37236/1481 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Niklas Eriksen ◽  
Henrik Eriksson ◽  
Kimmo Eriksson
Keyword(s):  
Golden Ratio ◽  
Upper Bounds ◽  
Combinatorial Interpretation ◽  
Eulerian Numbers ◽  
Signed Permutations

We introduce color-signed permutations to obtain a very explicit combinatorial interpretation of the $q$-Eulerian identities of Brenti and some generalizations. In particular, we prove an identity involving the golden ratio, which allows us to compute upper bounds on how high a checker can reach in a classical checker-jumping problem, when the rules are relaxed to allow also diagonal jumps.

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