Impulse Propagation in Compositions and Words

Margaret Archibald,Arnold Knopfmacher,Charlotte Brennan,Toufik Mansour,Aubrey Blecher,Sergejs Solovjovs

doi:10.1155/2021/8811261

Abstract

We consider compositions of n represented as bargraphs and subject these to repeated impulses which start from the left at the top level and destroy horizontally connected parts. This is repeated while moving to the right first and then downwards to the next row and the statistic of interest is the number of impulses needed to annihilate the whole composition. We achieve this by conceptualizing a generating function that tracks compositions as well as the number of impulses used. This conceptualization is repeated for words (over a finite alphabet) represented by bargraphs.

Highlights

In several recent papers modelling physical or chemical situations, molecules and other structures are represented by combinatorial objects and various physical entities acting on the objects are modelled in the mathematical context of these combinatorial objects
Bargraphs are nonintersecting lattice paths in N20 with three allowed types of steps; up (0, 1), down (0, − 1), and horizontal (1, 0). ey start at the origin with an up step and terminate immediately upon return to the x-axis
Bargraphs can represent compositions and words where the respective height of each horizontal step corresponds to the respective size of each part of the three structures

Summary

Introduction

In several recent papers modelling physical or chemical situations, molecules and other structures are represented by combinatorial objects (e.g., compositions) and various physical entities (such as light or force) acting on the objects are modelled in the mathematical context of these combinatorial objects (see [1,2,3,4]). An impulse starts from the left at the top level and destroys the blocks formed by the horizontally connected cells. Bargraphs can represent compositions and words where the respective height of each horizontal step corresponds to the respective size of each part of the three structures. We devote one section to each of the combinatorial structures (compositions and words). Each section develops the generating function for the number of impulses required to annihilate the respective structures, and we obtain the average number of such impulses required for each of the structures in relation to its defining parameter (size of the composition and length of words).

Compositions

Impulses at Level j