The inner site-perimeter of compositions

Abstract

Compositions of n are finite sequences of positive integers (σ_i)^k_i=1 such that

                                      σ₁ + σ₂ + ^{, , ,} + σ_k = n.

The σ's are called parts. We can represent a composition as a bargraph where the parts of the composition are represented by the columns and the height of each column corresponds to the size of the corresponding part. We consider the inner site-perimeter which is the total number of cells inside the bargraph that have at least one edge in common with an outside cell. The generating function that counts the inner site-perimeter of compositions is obtained. From this we nd the average inner site-perimeter and an asymptotic expression for this average as the size of the composition tends to innity. Finally we discuss the notion of a hole in a composition and count compositions with no holes.

Mathematics Subject Classification (2010): Primary: 05A15, 05A16; Secondary: 60C05.

Keywords: Bargraphs, site-perimeter, inner site-perimeter, generating functions, asymptotics