combinatorial identity
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2021 ◽  
Vol 344 (4) ◽  
pp. 112270
Author(s):  
Tony Dorlas ◽  
Alexei Rebenko ◽  
Baptiste Savoie

Mathematika ◽  
2021 ◽  
Vol 67 (2) ◽  
pp. 498-513
Author(s):  
Fan Ge ◽  
Gongxiang Liu

2020 ◽  
Author(s):  
Sumit Kumar Jha

Let $\sum_{d|n}d$ denote the sum of divisors of a positive integer $n$, and let $\prod_{j=1}^{\infty}(1-q^{j})^{r}=\sum_{n=0}^{\infty}p_{r}(n)q^{n}$. The aim is of this note is to prove the following interesting combinatorial identity$$\sum_{d|n}d=n\,\sum_{r=1}^{n}\frac{(-1)^{r}}{r}\,\binom{n}{r}\, p_{r}(n).$$


2020 ◽  
Vol 306 (1) ◽  
pp. 375-383
Author(s):  
Oksana Yakimova

Author(s):  
Yilmaz Simsek

The aim of this paper is to define new families of combinatorial numbers and polynomials associated with Peters polynomials. These families are also a modification of the special numbers and polynomials in [11]. Some fundamental properties of these polynomials and numbers are given. Moreover, a combinatorial identity, which calculates the Fibonacci numbers with the aid of binomial coefficients and which was proved by Lucas in 1876, is proved by different method with the help of these combinatorial numbers. Consequently, by using the same method, we give a new recurrence formula for the Fibonacci numbers and Lucas numbers. Finally, relations between these combinatorial numbers and polynomials with their generating functions and other well-known special polynomials and numbers are given.


2019 ◽  
Vol 94 (2) ◽  
pp. 253-257
Author(s):  
Benjamin Hackl

2018 ◽  
Vol 28 (4) ◽  
pp. 485-518
Author(s):  
MARTIN AUMÜLLER ◽  
MARTIN DIETZFELBINGER ◽  
CLEMENS HEUBERGER ◽  
DANIEL KRENN ◽  
HELMUT PRODINGER

We present an average-case analysis of a variant of dual-pivot quicksort. We show that the algorithmic partitioning strategy used is optimal, that is, it minimizes the expected number of key comparisons. For the analysis, we calculate the expected number of comparisons exactly as well as asymptotically; in particular, we provide exact expressions for the linear, logarithmic and constant terms.An essential step is the analysis of zeros of lattice paths in a certain probability model. Along the way a combinatorial identity is proved.


2017 ◽  
Vol 4 (4) ◽  
pp. 453-477 ◽  
Author(s):  
Sunil Chhita ◽  
Patrik Ferrari

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