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.
The main result of this paper is the generalization and proof of a conjecture by Gould and Quaintance on the asymptotic behavior of certain sequences related to the Bell numbers. Thereafter we show some applications of the main theorem to statistics of partitions of a finite set S, i.e., collections B 1 , B 2 , . . . , B k of non-empty disjoint subsets of S such that k i=1 B i = S, as well as to certain classes of partitions of [n].
A word over an alphabet [k] can be represented by a bargraph, where the
height of the i-th column is the size of the i-th part. If North is in the
direction of the positive y-axis and East is in the direction of the positive
x-axis, a light source projects parallel rays from the North-West direction,
at an angle of 45 degrees to the y-axis. These rays strike the cells of the
bargraph. We say a cell is lit if the rays strike its West facing edge or
North facing edge or both. With the use of matrix algebra we find the
generating function that counts the number of lit cells. From this we find
the average number of lit cells in a word of length n.
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