# The site-perimeter of words

Document Type: Research Paper

Authors

1 University of the Witwatersrand

2 1 Jan Smuts Avenue

Abstract

We define $[k]=\{1‎, ‎2‎, ‎3,\ldots,k\}$ to be a (totally ordered) {\em alphabet} on $k$ letters‎. ‎A {\em word} $w$ of length $n$ on the alphabet $[k]$ is an element of $[k]^n$‎. ‎A word can be represented by a bargraph which is a family of column-convex polyominoes whose lower edge lies on the $x$-axis and in which the height of the $i$-th column in the bargraph equals the size of the $i$-th part of the word‎. ‎Thus these bargraphs have heights which are less than or equal to $k$‎. ‎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 site-perimeter of words is obtained explicitly‎. ‎From a functional equation we find the average site-perimeter of words of length $n$ over the alphabet $[k]$‎. ‎We also show how these statistics may be obtained using a direct counting method and obtain the minimum and maximum values of the site-perimeters‎.

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