A combinatorial identity for the sum of divisors function involving $p_{r}(n)$

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$$\sum_{d|n}d=n\,\sum_{r=1}^{n}\frac{(-1)^{r}}{r}\,\binom{n}{r}\, p_{r}(n).$$

A new family of combinatorial numbers and polynomials associated with peters numbers and polynomials

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.

Dual-Pivot Quicksort: Optimality, Analysis and Zeros of Associated Lattice Paths

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.