Protected Cells in Compositions

Compositions (ordered partitions) of n are finite sequences of positive integers that sum to n. We represent a composition of n as a bargraph with area n such that the height of the i-th column of the bargraph equals the size of the i-th part of the composition. We consider the concept of protected cells and protected columns in the bargraph representation of the composition. An r-protected cell is a cell in which the shortest path to the outside has at least r+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r+1$$\end{document} steps (up, down, left or right). We obtain the average number of r-protected cells and protected columns. Finally we study the total protection number of a composition and compute the mean of this quantity over all compositions of n. We define the total protection number of a composition π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document} to be the sum of the protection numbers of each individual cell in that composition.