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.

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 The Skew and Relative Derangements of Type $B$

10.37236/1025 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
William Y.C. Chen ◽  
Jessica C.Y. Zhang
Keyword(s):  
Intermediate Structure ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Signed Permutations

By introducing the notion of relative derangements of type $B$, also called signed relative derangements, which are defined in terms of signed permutations, we obtain a type $B$ analogue of the well-known relation between the relative derangements and the classical derangements. While this fact can be proved by using the principle of inclusion and exclusion, we present a combinatorial interpretation with the aid of the intermediate structure of signed skew derangements.

scholarly journals Mixed Volumes of Hypersimplices

10.37236/5770 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Mixed Volumes ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Set Of Permutations

In this paper 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.

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 Upper Bounds for Stern's Diatomic Sequence and Related Sequences

10.37236/5342 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Colin Defant
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Upper Bounds ◽  
Combinatorial Interpretation ◽  
Powers Of 2 ◽  
Stern Sequence ◽  
Special Case

Let $(s_2(n))_{n=0}^\infty$ denote Stern's diatomic sequence. For $n\geq 2$, we may view $s_2(n)$ as the number of partitions of $n-1$ into powers of $2$ with each part occurring at most twice. More generally, for integers $b,n\geq 2$, let $s_b(n)$ denote the number of partitions of $n-1$ into powers of $b$ with each part occurring at most $b$ times. Using this combinatorial interpretation of the sequences $s_b(n)$, we use the transfer-matrix method to develop a means of calculating $s_b(n)$ for certain values of $n$. This then allows us to derive upper bounds for $s_b(n)$ for certain values of $n$. In the special case $b=2$, our bounds improve upon the current upper bounds for the Stern sequence. In addition, we are able to prove that $\displaystyle{\limsup_{n\rightarrow\infty}\frac{s_b(n)}{n^{\log_b\phi}}=\frac{(b^2-1)^{\log_b\phi}}{\sqrt 5}}$.

scholarly journals Calcul Basique des Permutations Signées, II: Analogues Finis des Fonctions de BesseL

10.37236/1324 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
Dominique Foata ◽  
Guo-Niu Han
Keyword(s):  
Bessel Functions ◽  
Combinatorial Interpretation ◽  
Signed Permutations ◽  
Basic Calculus ◽  
Permutation Statistic

The traditional basic calculus on permutation statistic distributions is extended to the case of signed permutations. This provides with a combinatorial interpretation of the basic Bessel functions and their finite analogues. Le calcul basique classique sur les distributions des statistiques des permutations est prolongé au cas des permutations signées. Ce calcul permet ainsi de donner une interprétation combinatoire aux fonctions basiques de Bessel et à leurs analogues finis.

scholarly journals Generalized Stirling Permutations and Forests: Higher-Order Eulerian and Ward Numbers

10.37236/4814 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
J. Fernando Barbero G. ◽  
Jesús Salas ◽  
Eduardo J.S. Villaseñor
Keyword(s):  
Generating Function ◽  
Higher Order ◽  
Fixed Number ◽  
Combinatorial Interpretation ◽  
Eulerian Numbers ◽  
New Class ◽  
Ordered Trees ◽  
New Family

We define a new family of generalized Stirling permutations that can be interpreted in terms of ordered trees and forests. We prove that the number of generalized Stirling permutations with a fixed number of ascents is given by a natural three-parameter generalization of the well-known Eulerian numbers. We give the generating function for this new class of numbers and, in the simplest cases, we find closed formulas for them and the corresponding row polynomials. By using a non-trivial involution our generalized Eulerian numbers can be mapped onto a family of generalized Ward numbers, forming a Riordan inverse pair, for which we also provide a combinatorial interpretation.

scholarly journals Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences

10.37236/1453 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Michael E. Hoffman
Keyword(s):  
Generating Functions ◽  
Root Systems ◽  
Class Numbers ◽  
Euler Polynomials ◽  
Combinatorial Interpretation ◽  
Integer Sequences ◽  
Rational Argument ◽  
Signed Permutations ◽  
Secant Numbers

Let $P_n$ and $Q_n$ be the polynomials obtained by repeated differentiation of the tangent and secant functions respectively. From the exponential generating functions of these polynomials we develop relations among their values, which are then applied to various numerical sequences which occur as values of the $P_n$ and $Q_n$. For example, $P_n(0)$ and $Q_n(0)$ are respectively the $n$th tangent and secant numbers, while $P_n(0)+Q_n(0)$ is the $n$th André number. The André numbers, along with the numbers $Q_n(1)$ and $P_n(1)-Q_n(1)$, are the Springer numbers of root systems of types $A_n$, $B_n$, and $D_n$ respectively, or alternatively (following V. I. Arnol'd) count the number of "snakes" of these types. We prove this for the latter two cases using combinatorial arguments. We relate the values of $P_n$ and $Q_n$ at $\sqrt3$ to certain "generalized Euler and class numbers" of D. Shanks, which have a combinatorial interpretation in terms of 3-signed permutations as defined by R. Ehrenborg and M. A. Readdy. Finally, we express the values of Euler polynomials at any rational argument in terms of $P_n$ and $Q_n$, and from this deduce formulas for Springer and Shanks numbers in terms of Euler polynomials.

scholarly journals Crossings of signed permutations and $q$-Eulerian numbers of type $B$

2013 ◽  
Vol 4 (2) ◽  
pp. 191-228 ◽  
Author(s):  
Sylvie Corteel ◽  
Matthieu Josuat-Vergès ◽  
Jang Soo Kim
Keyword(s):  
Type B ◽  
Eulerian Numbers ◽  
Signed Permutations
Sign in / Sign up

Export Citation Format

Share Document