It is known that the integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. In this paper, we give an efficient data structure for storing all ascending compositions of a positive integer. Using this structure, we improved the fastest known algorithm for generating integer partitions.

In this note we introduce a new method to proving and discovering some identities involving binomial coefficients and factorials.

A new expansion is given for partial sums of Euler's pentagonal number series. As a corollary we derive an infinite family of inequalities for the partition function, p(n).

Using the multisection series method, we establish formulas for various power sums of cosine functions. As corollaries we derive several combinatorial identities.

In this paper we give a fast algorithm to generate all partitions of a positive integer n. Integer partitions may be encoded as either ascending or descending compositions for the purposes of systematic generation. It is known that the ascending composition generation algorithm is substantially more efficient than its descending composition counterpart. Using tree structures for storing the partitions of integers, we develop a new ascending composition

The aim of this paper is to present a method of generating inequalities and, under certain conditions, some identities with sums that involve floor, ceiling and round functions. We apply this method to sequences of nonnegative integers that could be turned into periodical sequences.