Abstract:
Compositions of $n$ are finite sequences of positive integers $(\sigma_i)_{i=1}^k$ such that \[\sigma_1+\sigma_2+\cdots +\sigma_k=n.\] We represent a composition of $n$ as a bargraph with area $n$ such that the height of the $i$-th column of the bargraph equals the size of the $i$-th part of the composition. We consider the site-perimeter which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the total site-perimeter of compositions is obtained. In addition, we rederive the average site-perimeter of a composition by direct counting. Finally we determine the average site-perimeter of a bargraph with a given semi-perimeter.
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