arXiv subjects

Naomi Krawzik

Publications and source records attributed to Naomi Krawzik.

5 recordsLinked to original sources

arXiv2025

The wild number of an edge-colored graph

We introduce the wild number of an edge-colored graph as a measure of how close an edge-colored graph is to having a spanning tree in every color. This combinatorial concept originates in the algebraic theory of generalized graph splines. After showing that determining the wild number of a graph is an NP-complete problem, we provide bounds on the wild number and find the exact wild number for trees, cycles, and families of graphs with restrictions on the edge-colorings. This article serves as an invitation to the topic of wild numbers and includes several open problems, many of which are suitable for undergraduate research projects.

math.CO↗

arXiv2024

Graded deformations of skew group algebras for cyclic transvection groups acting on polynomial rings in positive characteristic

We investigate deformations of skew group algebras that arise from a finite cyclic group acting on a polynomial ring in positive characteristic, where characteristic divides the order of the group. We allow deformations which deform both the group action and the vector space multiplication. We fully characterize the Poincare-Birkhoff-Witt deformations which arise in this setting from multiple perspectives: a necessary and sufficient condition list, a practical road map from which one can generate examples corresponding to any choice of group algebra element, an explicit formula, and a combinatorial analysis of the class of algebras.

math.RA↗

arXiv2024

Primeness of generalized parking functions

Classical parking functions are a generalization of permutations that appear in many combinatorial structures. Prime parking functions are indecomposable components such that any classical parking function can be uniquely described as a direct sum of prime ones. In this article, we extend the notion of primeness to three generalizations of classical parking functions: vector parking functions, $(p,q)$-parking functions, and two-dimensional vector parking functions. We study their enumeration by obtaining explicit formulas for the number of prime vector parking functions when the vector is an arithmetic progression, prime $(p,q)$-parking functions, and prime two-dimensional vector parking functions when the weight matrix is an affine transformation of the coordinates.

math.CO↗

arXiv2024

Maximum likelihood degree of the $\beta$-stochastic blockmodel

Log-linear exponential random graph models are a specific class of statistical network models that have a log-linear representation. This class includes many stochastic blockmodel variants. In this paper, we focus on $\beta$-stochastic blockmodels, which combine the $\beta$-model with a stochastic blockmodel. Here, using recent results by Almendra-Hern\'{a}ndez, De Loera, and Petrovi\'{c}, which describe a Markov basis for $\beta$-stochastic block model, we give a closed form formula for the maximum likelihood degree of a $\beta$-stochastic blockmodel. The maximum likelihood degree is the number of complex solutions to the likelihood equations. In the case of the $\beta$-stochastic blockmodel, the maximum likelihood degree factors into a product of Eulerian numbers.

math.ST↗

arXiv2022

Drinfeld Hecke algebras for symmetric groups in positive characteristic

We investigate deformations of skew group algebras arising from the action of the symmetric group on polynomial rings over fields of arbitrary characteristic. Over the real or complex numbers, Lusztig's graded affine Hecke algebra and analogs are all isomorphic to Drinfeld Hecke algebras, which include the symplectic reflection algebras and rational Cherednik algebras. Over fields of prime characteristic, new deformations arise that capture both a disruption of the group action and also a disruption of the commutativity relations defining the polynomial ring. We classify deformations for the symmetric group acting in its natural (reducible) reflection representation.

math.RT↗