@article{tjant2019742,
author={{Mansour, Toufik and Shattuck, Mark},
title={Counting Water Cells in Pattern Restricted Compositions},
journal={Turkish Journal of Analysis and Number Theory},
volume={7},
number={4},
pages={98--112},
year={2019},
url={http://pubs.sciepub.com/tjant/7/4/2},
issn={2333-1232},
abstract={In this paper, we consider statistics on compositions of a positive integer represented geometrically as bargraphs that avoid certain classes of consecutive patterns. A unit square exterior to a bargraph that lies along a horizontal line between any two squares contained within its subtended area is called a <i>water cell</i> since it is a place where a liquid would collect if poured along the top part of the bargraph from above. The total number of water cells in the bargraph representation of a k-ary word then gives what is referred to as the <i>capacity of</i> w. Here, we determine the distribution of the capacity statistic on certain pattern-restricted compositions, regarded as k-ary words. Several general classes of patterns are considered, including <img src=image/abs1.png></img> and <img src=image/abs2.png></img> where a is arbitrary. As a consequence of our results, we obtain all of the distinct distributions for the capacity statistic on avoidance classes of compositions corresponding to 3-letter patterns having at most two distinct letters. Finally, in the case of <img src=image/abs3.png></img> some further enumerative results are given when a=2, including algebraic and bijective proofs for the total capacity of all Carlitz partitions of a given size having a fixed number of blocks.},
doi={10.12691/tjant-7-4-2}
publisher={Science and Education Publishing}
}