Compositions of Positive Integers \(n\) Viewed as Alternating Sequences of Increasing/Decreasing Partitions

Abstract

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by \(q\)-series and compositions exhibiting particular patterns are specified by generating functions for these patterns. Here we view compositions as alternating sequences of partitions (i.e., alternating blocks) and obtain results for the asymptotic expectations of the number of such blocks (or parts per block) for different ways of defining the blocks.