# Marcel Kieren GohMcGill University | McGill · Department of Mathematics and Statistics

Mathematics

12
Publications
390
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3
Citations
Introduction
I'm interested in the analysis of random combinatorial objects, analytic combinatorics, as well as additive combinatorics.
Skills and Expertise
Education
September 2017 - April 2021
Field of study
• Honours Mathematics and Computer Science

## Publications

Publications (12)
Preprint
Full-text available
We introduce a class of set families that includes the collection of primitive sets, pairwise coprime sets, and product-free sets. If F is a set family in our class, we let F n,k be the number of elements in F ∩ 2 {1,2,...,n} with cardinality exactly k and show that n k=0 (−1) k F n,k = K F , where K F is a constant depending on the family F but no...
Article
We study several parameters of a random Bienaymé–Galton–Watson tree $T_n$ of size $n$ defined in terms of an offspring distribution $\xi$ with mean $1$ and nonzero finite variance $\sigma ^2$ . Let $f(s)=\mathbb{E}\{s^\xi \}$ be the generating function of the random variable $\xi$ . We show that the independence number is in probability asymptotic...
Article
Full-text available
This note defines a notion of multiplicity for nodes in a rooted tree and presents an asymptotic calculation of the maximum multiplicity over all leaves in a Bienaymé-Galton-Watson tree with critical offspring distribution ξ, conditioned on the tree being of size n. In particular, we show that if Sn is the maximum multiplicity in a conditioned Bien...
Article
Full-text available
Given only the free-tree structure of a tree, the root estimation problem asks if one can guess which of the free tree's nodes is the root of the original tree. We determine the maximum-likelihood estimator for the root of a free tree when the underlying tree is a size-conditioned Galton–Watson tree and calculate its probability of being correct.
Article
Full-text available
For a finite set A of size n, an ordering is an injection from {1, 2,. .. , n} to A. We present results concerning the asymptotic properties of the length L n of the longest arithmetic subsequence in a random ordering of an additive set A. In the torsion-free case where A = [1, n] d ⊆ Z d , we prove that L n ∼ 2d log n/ log log n. We show that the...
Preprint
Full-text available
We study several parameters of a random Bienaymé-Galton-Watson tree Tn of size n defined in terms of an offspring distribution ξ with mean 1 and nonzero finite variance σ 2. Let f (s) = E{s ξ } be the generating function of the random variable ξ. We show that the independence number is in probability asymptotic to qn, where q is the unique solution...
Preprint
Full-text available
In this paper we investigate properties of the lattice $L_n$ of subsets of $[n] = \{1,\ldots,n\}$ that are arithmetic progressions, under the inclusion order. For $n\geq 4$, this poset is not graded and thus not semimodular. We start by deriving properties of the number $p_{nk}$ of arithmetic progressions of length $k$ in $[n]$. Next, we look at th...
Preprint
Full-text available
This note defines a notion of multiplicity for nodes in a rooted tree and presents an asymptotic calculation of the maximum multiplicity over all leaves in a Bienaym\'e-Galton-Watson tree with critical offspring distribution $\xi$, conditioned on the tree being of size $n$. In particular, we show that if $S_n$ is the maximum multiplicity in a condi...
Technical Report
Full-text available
We describe the results of a semi-computational search for regularity in Tlingit verb prefix charts. We present a set of twenty-eight rewrite rules that underlie phonological and morphological changes in the verb, and give an explicit sequence of rewrite rules that resolves every entry in the charts.
Preprint
Full-text available
For a finite set $A$ of size $n$, an ordering is an injection from $\{1,2,\ldots,n\}$ to $A$. We present results concerning the asymptotic properties of the length $L_n$ of the longest arithmetic subsequence in a random ordering of an additive set $A$. In the torsion-free case where $A = [1,n]^d\subseteq {\bf Z}^d$, we prove that there exists a fun...
Preprint
Full-text available
Given only the free-tree structure of a tree, the root estimation problem asks if one can guess which of the free tree's nodes is the root of the original tree. We determine the maximum-likelihood estimator for the root of a free tree when the underlying tree is a size-conditioned Galton-Watson tree and calculate its probability of being correct.
Technical Report
Full-text available
The vast majority of software contains bugs, and various methods have been devised to find bugs and prevent their creation. Formalising programming languages and proving theorems about them is one way of verifying the soundness of programs. Proof assistants provide an interactive medium for constructing such proofs and they are widely used in progr...

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