scholarly journals Some combinatorial identities of the r-Whitney-Eulerian numbers

2019 ◽  
Vol 13 (2) ◽  
pp. 378-398
Author(s):  
Toufik Mansour ◽  
José Ramírez ◽  
Mark Shattuck ◽  
Sergio Villamaríín
Keyword(s):  
Generating Function ◽  
Combinatorial Identities ◽  
Lambert W Function ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Exponential Generating Function

In this paper, we study further properties of a recently introduced generalized Eulerian number, denoted by Am,r(n, k), which reduces to the classical Eulerian number when m = 1 and r = 0. Among our results is a generalization of an earlier symmetric Eulerian number identity of Chung, Graham and Knuth. Using the row generating function for Am,r(n, k) for a fixed n, we introduce the r-Whitney-Euler-Frobenius fractions, which generalize the Euler-Frobenius fractions. Finally, we consider a further four-parameter combinatorial generalization of Am,r(n, k) and find a formula for its exponential generating function in terms of the Lambert-W function.

scholarly journals IDENTITIES OF SYMMETRY FOR GENERALIZED TWISTED BERNOULLI POLYNOMIALS TWISTED BY RAMIFIED ROOTS OF UNITY

2014 ◽  
Vol 60 (1) ◽  
pp. 19-36
Author(s):  
Dae San Kim
Keyword(s):  
Generating Function ◽  
Bernoulli Polynomials ◽  
Roots Of Unity ◽  
Integral Expression ◽  
Power Sums ◽  
Exponential Generating Function

Abstract We derive eight identities of symmetry in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by ramified roots of unity. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.

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 $D$.

A generalized class of restricted Stirling and Lah numbers

2018 ◽  
Vol 68 (4) ◽  
pp. 727-740 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Combinatorial Properties ◽  
Exponential Generating Function ◽  
Cauchy Numbers ◽  
Restricted Stirling Numbers

Abstract In this paper, we consider a polynomial generalization, denoted by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) as well as for the associated Cauchy numbers.

Random Graphs from a Minor-Closed Class

2009 ◽  
Vol 18 (4) ◽  
pp. 583-599 ◽  
Author(s):  
COLIN McDIARMID
Keyword(s):  
Poisson Distribution ◽  
Random Graph ◽  
Generating Function ◽  
Random Graphs ◽  
Giant Component ◽  
Closed Class ◽  
Exponential Generating Function ◽  
Excluded Minor ◽  
A Minor ◽  
Labelled Graphs

A minor-closed class of graphs is addable if each excluded minor is 2-connected. We see that such a classof labelled graphs has smooth growth; and, for the random graphRnsampled uniformly from then-vertex graphs in, the fragment not in the giant component asymptotically has a simple ‘Boltzmann Poisson distribution’. In particular, asn→ ∞ the probability thatRnis connected tends to 1/A(ρ), whereA(x) is the exponential generating function forand ρ is its radius of convergence.

Eulerian number asymptotics

1994 ◽  
Vol 445 (1924) ◽  
pp. 291-303 ◽  
Keyword(s):  
Formal Methods ◽  
Asymptotic Expansions ◽  
Recursion Relation ◽  
Matched Asymptotic Expansions ◽  
Ray Method ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Asymptotic Formulae

Asymptotic formulae for the eulerian numbers A(n, k) for n ≫ 1 are obtained directly from their recursion relation by the ray method and the method of matched asymptotic expansions. These are formal methods, so they do not prove that the formulae are asymptotic, although they suggest it. The formulae agree with the previously known results where those results are valid. They also agree very well with the exact values of A(n, k) for 1 ≼ n ≼ 170, and the agreement improves as n increases.

scholarly journals On some combinatorial identities and harmonic sums

2017 ◽  
Vol 13 (07) ◽  
pp. 1695-1709 ◽  
Author(s):  
Necdet Batir
Keyword(s):  
Integral Representation ◽  
Closed Form ◽  
Generating Function ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Generalized Harmonic Numbers

For any [Formula: see text] we first give new proofs for the following well-known combinatorial identities [Formula: see text] and [Formula: see text] and then we produce the generating function and an integral representation for [Formula: see text]. Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that [Formula: see text] and [Formula: see text] where [Formula: see text] are generalized harmonic numbers defined below.

scholarly journals On a New Family of Generalized Stirling and Bell Numbers

10.37236/564 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Toufik Mansour ◽  
Matthias Schork ◽  
Mark Shattuck
Keyword(s):  
Closed Form ◽  
Generating Function ◽  
Recursion Relation ◽  
Stirling Numbers ◽  
Combinatorial Interpretation ◽  
Bell Numbers ◽  
New Family ◽  
Exponential Generating Function ◽  
Rook Numbers ◽  
Generalized Stirling Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell 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 The Maximum Independent Sets of de Bruijn Graphs of Diameter 3

10.37236/681 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Dustin A. Cartwright ◽  
María Angélica Cueto ◽  
Enrique A. Tobis
Keyword(s):  
Recurrence Relation ◽  
Generating Function ◽  
Independent Sets ◽  
De Bruijn Graph ◽  
Alphabet Size ◽  
De Bruijn Graphs ◽  
Exponential Generating Function ◽  
De Bruijn

The nodes of the de Bruijn graph $B(d,3)$ consist of all strings of length $3$, taken from an alphabet of size $d$, with edges between words which are distinct substrings of a word of length $4$. We give an inductive characterization of the maximum independent sets of the de Bruijn graphs $B(d,3)$ and for the de Bruijn graph of diameter three with loops removed, for arbitrary alphabet size. We derive a recurrence relation and an exponential generating function for their number. This recurrence allows us to construct exponentially many comma-free codes of length 3 with maximal cardinality.

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