William M. Y. Goh

The number of distinct part sizes in a random integer partition

Abstract We prove a central limit theorem for the number of different part sizes in a random integer partition. If λ is one of the P(n) partitions of the integer n, let Dn(λ) be the number of distinct part sizes that λ has. (Each part size counts once, even though there may be many parts of a given size.) For any fixed x, #(λ: D n (λ) ⩽ A n + xB n } P(n) → 1 2π ∫ −∞ x l −t 2 2 dt as n → ∞, where A n = (√6/π)n 1 2 and B n = (ρ6/2π − √54/π 3 ) 1 2 n 1 4 .

A central limit theorem on gln(fq)

For T E GL,(F,), let Q , ( T ) be the number of irreducible factors that the characteristic polynomial of T has. We prove that, for any fixed x , # { T : R,(T)<log n + x ) / # G L , , ( F , ) + ( l / G ) j tX e-rz’2 as n + m .

Limit distribution for the maximum degree of a random recursive tree

If a recursive tree is selected uniformly at random from among all recursive trees on n vertices, then the distribution of the maximum in-degree Δ is given asymptotically by the following theorem: for any fixed integer d, Pn(Δ ≤ [µn] + d) = exp(-2{µn}-d-1) + o(1) as n → ∞ , where µn = log2n. (As usual, [µn] denotes the greatest integer less than or equal to µn, and {µn} = µn- [µn].) The proof makes extensive use of asymptotic approximations for the partial sums of the exponential series.

Note: Gaps in samples of geometric random variables

In this note we continue the study of gaps in samples of geometric random variables originated in Hitczenko and Knopfmacher [Gap-free compositions and gap-free samples of geometric random variables. Discrete Math. 294 (2005) 225-239] and continued in Louchard and Prodinger [The number of gaps in sequences of geometrically distributed random variables, Preprint available at (number 81 on the list) or at ] In particular, since the notion of a gap differs in these two papers, we derive some of the results obtained in Louchard and Prodinger [The number of gaps in sequences of geometrically distributed random variables, Preprint available at (number 81 on the list) or at ] for gaps as defined in Hitczenko and Knopfmacher [Gap-free compositions and gap-free samples of geometric random variables. Discrete Math. 294 (2005) 225-239].

Average Number of Distinct Part Sizes in a Random Carlitz Composition

A composition of an integer n is called Carlitz if adjacent parts are different. Several characteristics of random Carlitz compositions have been studied recently by Knopfmacher and Prodinger. We will complement their work by establishing asymptotics of the average number of distinct part sizes in a random Carlitz composition.

On the asymptotics of the Tricomi-Carlitz polynomials and their zero distribution (I)

The asymptotic behavior of the Tricomi–Carlitz polynomials in the complex plane is established.

The maximum degree in a random tree and related problems

Meir and Moon studied the distribution of the maximum degree for simply generated families of trees. We have sharper results for the special case of labelled trees.

Unlabeled trees: Distribution of the maximum degree

Pick a tree uniformly at random from among all unlabeled trees on n vertices, and let Xn be the maximum of the degrees of its vertices. For any fixed integer d, as n∞, where μn = c1 log n, where {μn} : = μn – ⌊μ⌋ denotes the fractional part of μn and where co, c1, and η∞ are knnown constants, givenb approximately by c0 = 3.262 …, c1 = 0.9227 …, and η∞ = 0.3383….

Gap-Free Set Partitions

A set partition is called “gap-free” if its block sizes form an interval. In other words, there is at least one block of each size between the smallest and largest block sizes. Let B(n) and G(n), respectively, denote the number of partitions and the number of gap-free partitions of the set [n]. We prove that ***image***

Random Set Partitions

For random partitions of