Marcel Kieren Goh

• Mathematics
• Graduate researcher at McGill University

This note derives asymptotic upper and lower bounds for the number of planted plane trees on $n$ nodes assigned labels from the set $\{1,2,\ldots, k\}$ with the restriction that on any path from the root to a leaf, the labels must strictly decrease. We illustrate an application to calculating the largest eigenvalue of the adjacency matrix of a tree...

This paper describes the homology of various simplicial complexes associated with set families from combinatorial number theory, including primitive sets, pairwise coprime sets, product-free sets, and coprime-free sets. We present a condition on a set family that results in easy computation of the homology groups and show that the first three examp...

We give conditions for a locally finite poset $P$ to have the property that for any functions $f:P\to {\bf C}$ and $g:P\to {\bf C}$ not identically zero and linked by the M\"obius inversion formula, the support of at least one of $f$ and $g$ is infinite. This generalises and gives an entirely poset-theoretic proof of a result of Pollack. Various ex...

In this paper we investigate properties of the lattice L n of subsets of [n] = {1,. .. , n} that are arithmetic progressions, under the inclusion order. For n ≥ 4, this poset is not graded and thus not semimodular. We give three independent proofs of the fact that for n ≥ 2, μ n (L n) = μ(n − 1), where μ n is the Möbius function of L n and μ is the...

This thesis presents three results concerning conditional Galton--Watson trees. Each of these results involves, in some way, the structure or shape of the underlying tree. First we tackle the root estimation problem in Galton--Watson trees, whose setup is as follows. A Galton--Watson tree is generated with a known offspring distribution, conditione...

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...

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...

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...

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.

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...

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...

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...

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...

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.

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...

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.

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...