Article
Combinatorial parameters in bargraphs
DOI:
10.2989/16073606.2015.1121932
Author(s):
Aubrey BlecherThe John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, South Africa, Charlotte BrennanThe John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, South Africa, Arnold KnopfmacherThe John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, South Africa,
Abstract
Bargraphs are non-intersecting lattice paths in with 3 allowed types of steps; up (0, 1), down (0, −1) and horizontal (1, 0). They start at the origin with an up step and terminate immediately upon return to the x-axis. We unify the study of integer compositions (ordered partitions) with that of bargraph lattice paths by obtaining a single generating function for both these structures. We also obtain the asymptotic expected size of the underlying composition associated with an arbitrary bargraph as the semiperimeter tends to infinity (equivalently the expected value for the total area under the bargraph). In addition, the number of descents, the number of up steps and the number of level steps are found together with their asymptotic expressions for large semiperimeter.
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