Abstract
A word over an alphabet [k] can be represented by a bargraph, where the height of the i-th column is the size of the i-th part. If North is in the direction of the positive y-axis and East is in the direction of the positive x-axis, a light source projects parallel rays from the North-West direction, at an angle of 45 degrees to the y-axis. These rays strike the cells of the bargraph. We say a cell is lit if the rays strike its West facing edge or North facing edge or both. With the use of matrix algebra we find the generating function that counts the number of lit cells. From this we find the average number of lit cells in a word of length n.
Highlights
We define [k] = {1, 2, 3, · · ·, k} to be a alphabet on k letters
A word can be represented by a bargraph which is a column-convex polyominoe whose lower edge lies on the x-axis and in which the height of the i-th column in the bargraph equals the number of cells in the corresponding part of that word
We let North be the direction of the positive y-axis and East the direction of the positive x axis
Summary
If North is in the direction of the positive y-axis and East is in the direction of the positive x-axis, a light source projects parallel rays from the North-West direction, at an angle of 45 degrees to the y-axis. These rays strike the cells of the bargraph. With the use of matrix algebra we find the generating function that counts the number of lit cells. From this we find the average number of lit cells in a word of length n
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