eng Science and Education Publishing Turkish Journal of Analysis and Number Theory 2333-1232 2019-07-24 7 4 98 112 10.12691/tjant-7-4-2 TJANT2019742 article Toufik Mansour tmansour@univ.haifa.ac.il 1 Mark Shattuck 2 Department of Mathematics, University of Haifa, 3498838 Haifa, Israel Department of Mathematics, University of Tennessee, 37996 Knoxville, TN, USA 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 water cell 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 capacity of 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 and 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 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. http://pubs.sciepub.com/tjant/7/4/2/tjant-7-4-2.pdf consecutive pattern combinatorial statistic pattern-avoiding composition Carlitz composition