On the Partitions of a Number into Arithmetic Progressions

On rainbow 4-term arithmetic progressions

{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...

Combinatorics of Integer Partitions in Arithmetic Progression

The partitions of a positive integer n in which the parts are in arithmetic progression possess interesting combinatorial properties that distinguish them from other classes of partitions. We exhibit the properties by analyzing partitions with respect to a fixed length of the arithmetic progressions. We also address an open question concerning the number of integers k for which there is a k-par...

Avoiding triples in arithmetic progression ∗

Some patterns cannot be avoided ad infinitum. A well-known example of such a pattern is an arithmetic progression in partitions of natural numbers. We observed that in order to avoid arithmetic progressions, other patterns emerge. A visualization is presented that reveals these patterns. We capitalize on the observed patterns by constructing techniques to avoid arithmetic progressions. More for...

On arithmetic partitions of Zn

Generalizing a classical problem in enumerative combinatorics, Mansour and Sun counted the number of subsets of Zn without certain separations. Chen, Wang, and Zhang then studied the problem of partitioning Zn into arithmetical progressions of a given type under some technical conditions. In this paper, we improve on their main theorems by applying a convolution formula for cyclic multinomial c...

Local distribution of the parts of unequal partitions in arithmetic progressions II