Levels in bargraphs

Abstract

Bargraphs are lattice paths in N 2 0 , which start at the origin and terminate immediately upon return to the x-axis. The allowed steps are the up step (0, 1), the down step (0, −1) and the horizontal step (1, 0). The first step is an up step and the horizontal steps must all lie above the x-axis. An up step cannot follow a down step and vice versa. In this paper we consider levels, which are maximal sequences of two or more adjacent horizontal steps. We find the generating functions that count the total number of levels, the leftmost x-coordinate and the height of the first level and obtain the generating function for the mean of these parameters. Finally, we obtain the asymptotics of these means as the length of the path tends to infinity.

Enumerations of bargraphs with respect to corner statistics

“…Let P k (x, y, q) be the generating function for the number of set partitions π of [n] with exactly k blocks according to the number of cells in π, the number of columns of π (which is n), and the number of (a, b)-corners of type A in π corresponding to the variables x, y and q = (q a,b ) a,b≥1 respectively. Note that each set partition with exactly k blocks can be decomposed as 1π (1) · · · kπ (k) such that π (j) is a word over alphabet [j]. Thus, by Theorem 2.4, we have the following result.…”

“…Note that each set partition with exactly k blocks can be decomposed as 1π (1) · · · kπ (k) such that π (j) is a word over alphabet [j]. Thus, by Theorem 3.3, we have the following result.…”

Statistics on bargraphs viewed as cornerless Motzkin paths

“…Recall that a Motzkin path is a lattice path with up steps (1,1), horizontal steps (1, 0) and down steps (1, −1) that starts at the origin, ends on the x-axis, and never goes below the x-axis. A peak in a Motzkin path is an up step followed by a down step, and a valley is a down step followed by an up step.…”

“…Define a horizontal segment to be a maximal sequence of consecutive H steps, and let hs be the statistic "number of horizontal segments," both on B and on M. Horizontal segments of length at least 2 are considered in [1], where they are called levels.…”

A bijection between bargraphs and Dyck paths

“…As a special class of convex polyominoes, they have been studied and enumerated by Bousquet-Mélou and Rechnitzer [6], Prellberg and Brak [10], and Feretić [8]. The distribution of several bargraph statistics has been given in a series of papers by Blecher, Brennan, Knopfmacher and Prodinger [1,2,3,4,5]. All of the above papers rely on a decomposition of bargraphs that is used to obtain equations satisfied by the corresponding generating functions.…”