Charlotte Brennan

Combinatorial parameters in bargraphs

Abstract Bargraphs are non-intersecting lattice paths in with 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 the x-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.

The Pills problem revisited

We revisit the pills problem proposed by Knuth and McCarthy. In a bottle there are m large pills and n small pills. The large pill is equivalent to two small pills. Every day a person chooses a pill at random. If a small pill is chosen, it is eaten up, if a large pill is chosen it is broken into two halves, one half is eaten and the other half which is now considered to be a small pill is returned to the bottle. How many pills are left, on average, when the last large pill has disappeared? We show how to compute the moments, in particular the variance, and then generalize the problem in various ways.

Peaks in bargraphs

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 step and 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.

Levels in bargraphs

Bargraphs are lattice paths in N_0^2, 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 first and last ascents of size d or more in samples of geometric random variables

We consider words or strings of characters a 1 a 2 a 3…a n of length n, where the letters ai ∈ N are independently generated with a geometric probability P{X = k} = pq k−1 where p + q = 1. Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if a i+1 ≥ a i + d. We study the average position, initial height and end height of the first and last ascents of size d or more in a random geometrically distributed word.

The height and width of bargraphs

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.

Compositions of n as alternating sequences of weakly increasing and strictly decreasing partitions

Compositions and partitions of positive integers are often studied in separate frameworks where partitions are given by q-series generating functions and compositions exhibiting specific patterns are designated by generating functions for these patterns. Here, we view compositions as alternating sequences of weakly increasing and strictly decreasing partitions (i.e. alternating blocks). We obtain generating functions for the number of such partitions in terms of the size of the composition, the number of parts and the total number of “valleys” and “peaks”. From this, we find the total number of “peaks” and “valleys” in the composition of n which have the mentioned pattern. We also obtain the generating function for compositions which split into just two partition blocks. Finally, we obtain the two generating functions for compositions of n that start either with a weakly increasing partition or a strictly decreasing partition.

SEPARATION OF THE MAXIMA IN SAMPLES OF GEOMETRIC RANDOM VARIABLES

d > 0, we study the probability that the distance between the consecutive maxima in these samples is at least d. We derive a probability generating function for such samples and from it we obtain an exact formula for the probability as a double sum. Using Rice’s method we obtain asymptotic estimates for these probabilities. As a consequence of these results, we determine the average minimum separation of the maxima, in a sample of n geometric random variables with at least two maxima.

Peaks and Valleys in Motzkin Paths

Abstract A Motzkin path is a non-negative lattice path in N2 starting at the origin, where only three types of steps are allowed: the diagonal up step (1, 1) called u, the diagonal down step (1, −1) called d and the horizontal step (1, 0) called h. We consider paths of size n, ending at the point (n, 0). A peak is defined to be a node between the following steps: uh, hh, hd and ud, and a valley is a node between dh, hh, hu and du. A sharp peak or a sharp valley is the node between the steps ud and du respectively. We also define a low peak to be a peak on level one and a low valley to be a valley on level zero. In this paper, we find the asymptotic expressions for the average number of peaks, sharp peaks, low peaks, low sharp peaks and similarly for valleys on Motzkin paths of size n as n →∞.

Visits to Level r by Dyck Paths

A Dyck path is a non-negative lattice path in