Casting light on integer compositions

Abstract

Integer compositions of n are viewed as bargraphs with n circular nodes or square cells in which the ith part of the composition \(x_i\) is given by the ith column of the bargraph with \(x_i\) nodes or cells. The sun is at infinity in the north west of our two dimensional model and each node/cell may or may not be lit depending on whether it stands in the shadow cast by another node/cell to its left. We study the number of lit nodes in an integer composition of n and later we modify this to yield the number of lit square cells. We then count the number of columns being lit which leads naturally to those cases where only the first column is lit. We prove the theorem that the generating function for the latter is the same as the generating function for compositions in which the first part is strictly smallest. This theorem has interesting q-series identities as corollaries which allow us to deduce in a simple way the asymptotics for both the number of lit nodes and columns as \(n \rightarrow \infty \).