arXiv subjects

Eli B. Dugan

Publications and source records attributed to Eli B. Dugan.

3 recordsLinked to original sources

arXiv2026

Polyomino Density

A de Bruijn polyomino has colored cells and includes exactly one instance of each possible coloring of another polyomino with those colors. This generalizes the notion of a de Bruijn sequence. We are interested in the de Bruijn polyominoes of minimum size. As a step toward finding these, we introduce the problem of finding the smallest polyominoes containing at least $N$ translated copies of the polyomino $p$, allowing overlaps. We say these are $(p,N)$-dense and have size $a_{p,N}$. For certain pairs of polyominoes $p$ and $q$ of equal size, we show there are polyomino transformations relating $a_{p,N}$ and $a_{q,N}$. Using these transformations, we identify classes of polyominoes which share the sequence $(a_{p,N})_{N=1}^\infty$. We give closed forms for $(a_{p,N})_{N=1}^\infty$ for most polyominoes with up to five cells. Leveraging these results we introduce novel de Bruijn polyominoes, as well as other colored polyforms with de Bruijn-like properties.

math.CO↗

arXiv2025

Completions of Extremely Noncatenary Noetherian UFDs

Let $T$ be a complete local ring. We present necessary and sufficient conditions for $T$ to be the completion of a local (Noetherian) unique factorization domain $A$ such that there exist height one prime ideals $\{J_k\}_{k = 1}^{\infty}$ of $A$ satisfying the following conditions: (1) $J_k = J_{\ell}$ if and only if $k = \ell$, (2) there exist positive integers $n \neq m$ such that for each $k \in \mathbb{N}$, there are two saturated chains of prime ideals of $A$ of the form $J_k \subsetneq J^{(1)}_{k,2} \subsetneq \cdots \subsetneq J^{(1)}_{k,n - 1} \subsetneq M$ and $J_k \subsetneq J^{(2)}_{k,2} \subsetneq \cdots \subsetneq J^{(2)}_{k,m - 1} \subsetneq M,$ where $M$ is the maximal ideal of $A$, and (3) the prime ideals from condition (2) satisfy $J^{(i)}_{k,a} = J^{(j)}_{\ell,b}$ if and only if $i = j$, $k = \ell$, and $a = b$. We also find sufficient conditions for $T$ to be the completion of a local (Noetherian) unique factorization domain $B$ such that $B/J$ is not catenary for all height one prime ideals $J$ of $B$.

math.AC↗

arXiv2024

Some asymptotic results on $p$-lengths of factorizations for numerical semigroups and arithmetical congruence monoids

A factorization of an element $x$ in a monoid $(M, \cdot)$ is an expression of the form $x = u_1^{z_1} \cdots u_k^{z_k}$ for irreducible elements $u_1, \ldots, u_k \in M$, and the length of such a factorization is $z_1 + \cdots + z_k$. We introduce the notion of $p$-length, a generalized notion of factorization length obtained from the $\ell_p$-norm of the sequence $(z_1, \ldots, z_k)$, and present asymptotic results on extremal $p$-lengths of factorizations for large elements of numerical semigroups (additive submonoids of $\mathbb Z_{\ge 0}$) and arithmetical congruence monoids (certain multiplicative submonoids of $\mathbb Z_{\ge 1}$). Our results, inspired by analogous results for classical factorization length, demonstrate the types of combinatorial statements one may hope to obtain for sufficiently nice monoids, as well as the subtlety such asymptotic questions can have for general monoids.

math.AC↗