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We consider words or strings of characters a1a2a3 . . . an of length n, where the letters ai ∈ N are independently generated with a geometric probability P{X = k} = pq where p + q = 1. Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if ai+1 ≥ ai + d. We determine the mean, variance and limiting distribution of the… (More)

- Charlotte A. C. Brennan, Arnold Knopfmacher, Stephan G. Wagner
- Combinatorics, Probability & Computing
- 2008

A partition of a positive integer n is a finite sequence of positive integers a1, a2, . . . , ak such that a1 + a2 + · · · + ak = n and ai+1 ≥ ai for all i. We say n is the size of the partition, ai is the ith part of the partition and we call p(n) the number of partitions of n. For instance the 11 partitions of n = 6 are 6, 15, 24, 33, 222, 123, 114, 1113,… (More)

- Charlotte A. C. Brennan, Arnold Knopfmacher
- Discrete Mathematics & Theoretical Computer…
- 2009

A composition of a positive integer n is a finite sequence of positive integers a1, a2, . . . , ak such that a1 + a2 + · · ·+ ak = n. Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if ai+1 ≥ ai +d. We determine the mean, variance and limiting distribution of the number of ascents of size d or more in the set of… (More)

- Charlotte A. C. Brennan
- Ars Comb.
- 2011

Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more at position i if ai+1 ≥ ai + d. In (3) the present authors found the distribution of the number of ascents of size d or more in compositions of n. In this paper we study various statistics relating to the first ascent of size d or more in compositions of n. In particular… (More)

- Aubrey Blecher, Charlotte A. C. Brennan, Arnold Knopfmacher
- Discrete Mathematics & Theoretical Computer…
- 2014

- Margaret Archibald, Aubrey Blecher, Charlotte A. C. Brennan, Arnold Knopfmacher
- Electr. J. Comb.
- 2015

In this paper, compositions of n are studied. These are sequences of positive integers (σi) k i=1 whose sum is n. We define a maximum to be a part which is greater than or equal to all other parts. We investigate the size of the descents immediately following any maximum and we focus particularly on the largest and average of these, obtaining the generating… (More)

Compositions are conceptualized as non alternating sequences of blocks of non-decreasing and strictly decreasing partitions. We find the generating function F (x, y, q) where x marks the size of the composition, y the number of parts and q the number of such partition blocks minus 1. We form these blocks starting on the left-hand-side of the composition and… (More)

- Margaret Archibald, Aubrey Blecher, Charlotte A. C. Brennan, Arnold Knopfmacher, Toufik Mansour
- Australasian J. Combinatorics
- 2016

In this paper we study the largest parts in integer partitions according to multiplicities and part sizes. Firstly we investigate various properties of the multiplicities of the largest parts. We then consider the sum of the m largest parts first as distinct parts and then including multiplicities. Finally, we find the generating function for the sum of the… (More)

- Aubrey Blecher, Charlotte A. C. Brennan, Arnold Knopfmacher, Helmut Prodinger
- Discrete Applied Mathematics
- 2015