Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
István Mező ◽  
Ayhan Dil
Keyword(s):  
Generating Function ◽  
Fibonacci Sequence ◽  
Stirling Numbers ◽  
Second Order ◽  
Seidel Method ◽  
Exponential Generating Function ◽  
Hyperharmonic Numbers ◽  
Recurrence Sequences

AbstractIn this paper we use the Euler-Seidel method for deriving new identities for hyperharmonic and r-Stirling numbers. The exponential generating function is determined for hyperharmonic numbers, which result is a generalization of Gosper’s identity. A classification of second order recurrence sequences is also given with the help of this method.

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.

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 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.

scholarly journals Distribution of some quadratic linear recurrence sequences modulo 1

2014 ◽  
Vol 30 (1) ◽  
pp. 79-86
Author(s):  
ARTURAS DUBICKAS ◽  
Keyword(s):  
Limit Point ◽  
Fibonacci Sequence ◽  
Second Order ◽  
Linear Recurrence ◽  
Order Linear ◽  
Recurrence Sequences

We show that if a is an even integer then for every ξ ∈ R the smallest limit point of the sequence ||ξan||∞n=1 does not exceed |a|/(2|a| + 2) and this bound is best possible in the sense that for some ξ this constant cannot be improved. Similar (best possible) bound is also obtained for the smallest limit point of the sequence ||ξxn||∞n=1, where (xn)∞n=1 satisfies the second order linear recurrence xn = axn−1 + bxn−2 with a, b ∈ N satisfying a > b. For the Fibonacci sequence (Fn)∞n=1 our result implies that supξ∈R lim infn→∞ ||ξFn|| = 1/5, and e.g., in case when a > 3 is an odd integer, b = 1 and θ := a/2 + p a 2/4 + 1 it shows that supξ∈R lim infn→∞ ||ξθn|| = (a − 1)/2a.

Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions

2013 ◽  
Vol 11 (5) ◽  
Author(s):  
István Mező
Keyword(s):  
Hypergeometric Function ◽  
Generating Function ◽  
Arithmetic Progressions ◽  
Harmonic Numbers ◽  
Bell Numbers ◽  
Exponential Generating Function ◽  
Hyperharmonic Numbers

AbstractThere is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijovic, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear.

scholarly journals On (q, r, w)-Stirling Numbers of the Second Kind

2017 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernstein Polynomials ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Bernoulli Numbers And Polynomials ◽  
Exponential Generating Function

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q, r, w)-Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q, r, w)-Stirling numbers of the second kind and q-Bernoulli polynomials in w.

Those Stirling Numbers Again

1961 ◽  
Vol 4 (2) ◽  
pp. 149-151 ◽  
Author(s):  
N. S. Mendelsohn
Keyword(s):  
Generating Function ◽  
Stirling Numbers ◽  
Combinatorial Analysis ◽  
Exponential Generating Function ◽  
Made In

In his book [1] Combinatorial Analysis, J. Riordan (p. 32) refers to the continual rediscovery of the Stirling numbers. The author of this note has been surprised on many occasions by the number of different environments in which these numbers make a natural appearance and, in fact, this article is concerned with just such an occurrence. The connection is made in a study of the exponential generating function of nr.

scholarly journals Morphological and Ultrastructural Characterization of Hemocytes in an Insect Model, the Hematophagous Dipetalogaster maxima (Hemiptera: Reduviidae)

Insects ◽  
2021 ◽  
Vol 12 (7) ◽  
pp. 640
Author(s):  
Natalia R. Moyetta ◽  
Fabián O. Ramos ◽  
Jimena Leyria ◽  
Lilián E. Canavoso ◽  
Leonardo L. Fruttero
Keyword(s):  
Cell Biology ◽  
Cell Types ◽  
Transmission Electron ◽  
Ultrastructural Characterization ◽  
Current Classification ◽  
Contrast Microscopy ◽  
First Time ◽  
Controversial Aspect

Hemocytes, the cells present in the hemolymph of insects and other invertebrates, perform several physiological functions, including innate immunity. The current classification of hemocyte types is based mostly on morphological features; however, divergences have emerged among specialists in triatomines, the insect vectors of Chagas’ disease (Hemiptera: Reduviidae). Here, we have combined technical approaches in order to characterize the hemocytes from fifth instar nymphs of the triatomine Dipetalogaster maxima. Moreover, in this work we describe, for the first time, the ultrastructural features of D. maxima hemocytes. Using phase contrast microscopy of fresh preparations, five hemocyte populations were identified and further characterized by immunofluorescence, flow cytometry and transmission electron microscopy. The plasmatocytes and the granulocytes were the most abundant cell types, although prohemocytes, adipohemocytes and oenocytes were also found. This work sheds light on a controversial aspect of triatomine cell biology and physiology setting the basis for future in-depth studies directed to address hemocyte classification using non-microscopy-based markers.

scholarly journals Semi-automated regional classification of the style of activity of slow rock-slope deformations using PS InSAR and SqueeSAR velocity data

Landslides ◽  
2021 ◽  
Author(s):  
Chiara Crippa ◽  
Elena Valbuzzi ◽  
Paolo Frattini ◽  
Giovanni B. Crosta ◽  
Margherita C. Spreafico ◽  
...  
Keyword(s):  
Rock Slope ◽  
Progressive Failure ◽  
Principal Component ◽  
Cost Effective ◽  
Displacement Rate ◽  
Machine Learning Classification ◽  
Management Perspective ◽  
Alpine Environments

AbstractLarge slow rock-slope deformations, including deep-seated gravitational slope deformations and large landslides, are widespread in alpine environments. They develop over thousands of years by progressive failure, resulting in slow movements that impact infrastructures and can eventually evolve into catastrophic rockslides. A robust characterization of their style of activity is thus required in a risk management perspective. We combine an original inventory of slow rock-slope deformations with different PS-InSAR and SqueeSAR datasets to develop a novel, semi-automated approach to characterize and classify 208 slow rock-slope deformations in Lombardia (Italian Central Alps) based on their displacement rate, kinematics, heterogeneity and morphometric expression. Through a peak analysis of displacement rate distributions, we characterize the segmentation of mapped landslides and highlight the occurrence of nested sectors with differential activity and displacement rates. Combining 2D decomposition of InSAR velocity vectors and machine learning classification, we develop an automatic approach to characterize the kinematics of each landslide. Then, we sequentially combine principal component and K-medoids cluster analyses to identify groups of slow rock-slope deformations with consistent styles of activity. Our methodology is readily applicable to different landslide datasets and provides an objective and cost-effective support to land planning and the prioritization of local-scale studies aimed at granting safety and infrastructure integrity.

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