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We obtain the generating function for the number of columns of fixed height r in a bargraph (classified according to semi-perimeter). As initial case for two distinct methods we first find the generating function for columns of height 1. Then using a first-return-to-level-1 decomposition, we obtain the rational function version of the continued fraction generating function which allows us to derive separate recursions for its numerator and denominator. This then allows us to get the asymptotic average number of columns for each r. We also obtain an equivalent generating function by exploiting a sequential decomposition for bargraphs in terms of columns of height r. [ABSTRACT FROM AUTHOR]

Adding a larger column at the end yields Graph HT ht R (s, p) counts all non-descending compositions. Thus Graph HT ht For an arbitrary composition that ends in a column of height I a i , consider how two columns of heights I b i then I c i may be appended to form a well (see Figure 2). [Extracted from the article]

Bargraphs are column convex polyominoes, where the lower edge lies on a horizontal axis. We consider the inner site-perimeter, which is the total number of cells inside the bargraph that have at least one edge in common with an outside cell and obtain the generating function that counts this statistic. From this we find the average inner perimeter and the asymptotic expression for this average as the semi-perimeter tends to infinity. We finally consider those bargraphs where the inner site-perimeter is exactly equal to the area of the bargraph. [ABSTRACT FROM AUTHOR]

Compositions of n are finite sequences of positive integers Such that σ1 + σ2 + · · · + σk = n. The σ's are called parts. We can represent a composition as a bargraph where the parts of the composition are represented by the columns and the height of each column corresponds to the size of the corresponding part. We consider the inner site-perimeter which is the total number of cells inside the bargraph that have at least one edge in common with an outside cell. The generating function that counts the inner site-perimeter of compositions is obtained. From this we find the average inner site-perimeter and an asymptotic expression for this average as the size of the composition tends to infinity. Finally we discuss the notion of a hole in a composition and count compositions with no holes. [ABSTRACT FROM AUTHOR]

We define [k] = 1, 2, 3, ... k} to be a (totally ordered) alphabet on k letters. A word w of length n on the alphabet [k] is an element of [k]n. A word can be represented by a bargraph which is a family of column-convex polyominoes whose lower edge lies on the x-axis and in which the height of the i-th column in the bargraph equals the size of the i-th part of the word. Thus these bargraphs have heights which are less than or equal to k. We consider the site-perimeter, which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the site-perimeter of words is obtained explicitly. From a functional equation we find the average site-perimeter of words of length n over the alphabet [k]. We also show how these statistics may be obtained using a direct counting method and obtain the minimum and maximum values of the site-perimeters. [ABSTRACT FROM AUTHOR]

Bargraphs are lattice paths in N²0 with three 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. A wall of size r is a maximal sequence of r adjacent up steps. In this paper we develop the generating function for the total number of walls of fixed size r ≥ 1. We then derive asymptotic estimates for the mean number of such walls. [ABSTRACT FROM AUTHOR]

Bargraphs are non-intersecting lattice paths inwith 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 thex-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. [ABSTRACT FROM AUTHOR]

Bargraphs are specific polyominoes or lattice paths in. They start at the origin and end on the-axis. The allowed steps are the up step, the down stepand the horizontal step. There are a few restrictions: the first step has to be an up step and the horizontal steps must all lie above the-axis. An up step cannot follow a down step and vice versa. In this paper, we define peaks and consider various parameters relating to peaks. We find the generating functions that count these parameters and then find the mean for each statistic. We also compute the asymptotics of these means as the length of the semi-perimeter of the bargraph tends to infinity, where the semi-perimeter is the sum of all the up and horizontal steps. [ABSTRACT FROM AUTHOR]

We define [ k ] = { 1 , 2 , … , k } to be a (totally ordered) alphabet on k letters. A word w of length n on the alphabet [ k ] is an element of [ k ] n . A word can be represented by a bargraph (i.e., by a column-convex polyomino whose lower edges lie on the x -axis) in which the height of the i th column equals the size of the i th part of the word. Thus these bargraphs have heights which are less than or equal to k . We consider the perimeter, which is the number of edges on the boundary of the bargraph. By way of Cramer’s method and the kernel method, we obtain the generating function that counts the perimeter of words. Using these generating functions we find the average perimeter of words of length n over the alphabet [ k ] . We also show how the mean and variance can be obtained using a direct counting method. [ABSTRACT FROM AUTHOR]

A bargraph is a lattice path in N 0 2 with three allowed steps: the up step u = ( 0 , 1 ) , the down step d = ( 0 , − 1 ) and the horizontal step h = ( 1 , 0 ) . It starts at the origin with an up step and terminates as soon as it intersects the x -axis again. A down step cannot follow an up step and vice versa. The height of a bargraph is the maximum y coordinate attained by the graph. The width is the horizontal distance from the origin till the end. For bargraphs of fixed semi-perimeter n we find the generating functions for the total height and the total width and hence find asymptotic estimates for the average height and the average width. Our methodology makes use of a bijection between bargraphs and u u d d -avoiding Dyck paths. [ABSTRACT FROM AUTHOR]