Large qq convergence of random characteristic polynomials to random permutations and its applications

We extend an observation due to Stong that the distribution of the number of degree $d$ irreducible factors of the characteristic polynomial of a random $n \times n$ matrix over a finite field $\mathbb{F}_{q}$ converges to the distribution of the number of length $d$ cycles of a random permutation in $S_{n}$, as $q \rightarrow \infty$, by having any finitely many choices of $d$, say $d_{1}, \dots, d_{r}$. This generalized convergence will be used for the following two applications: the distribution of the cokernel of an $n \times n$ Haar-random $\mathbb{Z}_{p}$-matrix when $p \rightarrow \infty$ and a matrix version of Landau's theorem that estimates the number of irreducible factors of a random characteristic polynomial for large $n$ when $q \rightarrow \infty$.Comment: 18 pages. Comments are welcome

On integer partitions corresponding to numerical semigroups

Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical semigroups with a focus on classifying the partitions that appear in the image of the injection from numerical semigroups. In particular, we count the number of partitions that correspond to numerical semigroups in terms of genus, Frobenius number, and multiplicity, with some restrictions

Enumerating Parking Completions Using Join and Split

Given a strictly increasing sequence t with entries from [n] := {1, . . . , n}, a parking completion is a sequence c with |t| + |c| = n and |{t ∈ t | t 6 i}| + |{c ∈ c | c 6 i}| > i for all i in [n]. We can think of t as a list of spots already taken in a street with n parking spots and c as a list of parking preferences where the i-th car attempts to park in the ci-th spot and if not available then proceeds up the street to find the next available spot, if any. A parking completion corresponds to a set of preferences c where all cars park. We relate parking completions to enumerating restricted lattice paths and give formulas for both the ordered and unordered variations of the problem by use of a pair of operations termed Join and Split. Our results give a new volume formula for most Pitman-Stanley polytopes, and enumerate the signature parking functions of Ceballos and Gonz´alez D’Le´on

Virmid: accurate detection of somatic mutations with sample impurity inference

Detection of somatic variation using sequence from disease-control matched data sets is a critical first step. In many cases including cancer, however, it is hard to isolate pure disease tissue, and the impurity hinders accurate mutation analysis by disrupting overall allele frequencies. Here, we propose a new method, Virmid, that explicitly determines the level of impurity in the sample, and uses it for improved detection of somatic variation. Extensive tests on simulated and real sequencing data from breast cancer and hemimegalencephaly demonstrate the power of our model. A software implementation of our method is available at http://sourceforge.net/projects/virmid/

Counting core partitions and numerical semigroups using polytopes

A partition is an $a$-core partition if none of its hook lengths are divisible by $a$. It is well known that the number of $a$-core partitions is infinite and the number of simultaneous $(a, b)$-core partitions is a generalized Catalan number if $a$ and $b$ are relatively prime. In the first half of the dissertation, we give an expression for the number of simultaneous $(a_1,a_2,\dots, a_k)$-core partitions that is equal to the number of integer points in a polytope. In the second half, we discuss objects closely related to core partitions, called numerical semigroups, which are additive monoids that have finite complements in the set of non-negative integers. For a numerical semigroup $S$, the genus of $S$ is the number of elements in \NN \setminus S and the multiplicity is the smallest nonzero element in $S$. In 2008, Bras-Amor\'os conjectured that the number of numerical semigroups with genus $g$ is increasing as $g$ increases. Later, Kaplan posed a conjecture that implies Bras-Amor\'os conjecture. In this dissertation, we prove Kaplan's conjecture when the multiplicity is 4 or 6 by counting the number of integer points in a polytope. Moreover, we find a formula for the number of numerical semigroups with multiplicity 4 and genus $g$

Combinatorics on bounded free Motzkin paths and its applications

In this paper, we construct a bijection from a set of bounded free Motzkin paths to a set of bounded Motzkin prefixes that induces a bijection from a set of bounded free Dyck paths to a set of bounded Dyck prefixes. We also give bijections between a set of bounded cornerless Motzkin paths and a set of $t$-core partitions, and a set of bounded cornerless symmetric Motzkin paths and a set of self-conjugate $t$-core partitions. As an application, we get explicit formulas for the number of ordinary and self-conjugate $t$-core partitions with a fixed number of corners.Comment: 19 pages, 8 figures, 2 table