TY - JOUR

T1 - Counting Water Cells in Pattern Restricted Compositions

AU - Mansour, Toufik

AU - Shattuck, Mark

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

U2 - https://doi.org/10.12691/tjant-7-4-2

DO - https://doi.org/10.12691/tjant-7-4-2

M3 - مقالة

SN - 2333-1100

VL - 7

SP - 98

EP - 112

JO - Turkish Journal of Analysis and Number Theory

JF - Turkish Journal of Analysis and Number Theory

IS - 4

ER -