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Gabe Udell

Publications and source records attributed to Gabe Udell.

11 recordsLinked to original sources
arXiv2025

Inversions in parking functions

In this paper, we obtain a q-exponential generating function for inversions on parking functions via symmetric function theory and also through a direct bijection to rooted labeled forests. We then apply these techniques to unit interval parking functions to give analogous results. We conclude by introducing a probabilistic approach through which we obtain formulas for the total number of inversions and several other statistics across all parking functions and other sets of words closed under rearrangement.

math.CO↗
arXiv2025

Statistics on $\ell$-interval parking functions

The displacement of a car with respect to a parking function is the number of spots it must drive past its preferred spot in order to park. An $\ell$-interval parking function is one in which each car has displacement at most $\ell$. Among our results, we enumerate $\ell$-interval parking functions with respect to statistics such as inversion, displacement, and major index. We show that $1$-interval parking functions with fixed displacement exhibit a cyclic sieving phenomenon. We give closed formulas for the number of $1$-interval parking functions with a fixed number of inversions. We prove that a well-known bijection of Foata preserves the set of $\ell$-interval parking functions exactly when $\ell\leq 2$ or $\ell\geq n-2$, which implies that the inversion and major index statistics are equidistributed in these cases.

math.CO↗
arXiv2025

Factorization length distribution for affine semigroups V: explicit asymptotic behavior of weighted factorization lengths on numerical semigroups

We describe the asymptotic behavior of weighted factorization lengths on numerical semigroups. Our approach is geometric as opposed to analytic, explains the presence of Curry-Schoenberg B-splines as limiting distributions, and provides explicit error bounds (no implied constants left unspecified). Along the way, we explicitly bound the difference between the vector partition function and the number of integer points in the variable polytope for a $2 \times k$ matrix.

math.CO↗
arXiv2024

Characterizing finite groups whose enhanced power graphs have universal vertices

Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking $G\setminus\{1\}$ as the vertex set of $\Delta(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set consisting of the universal vertices of $\Delta(G)$ along the identity element. For a solvable group $G$, we present a necessary and sufficient conditon for $K(G)$ to be nontrivial. We also develop a connection between $\Delta(G)$ and $K(G)$ when $|G|$ is divisible by two distinct primes and the diameter of $\Delta(G)$ is $2$.

math.GR↗
arXiv2021

CR embeddability of quotients of the Rossi sphere via spectral theory

We look at the action of finite subgroups of $\operatorname{SU}(2)$ on $S^3$, viewed as a CR manifold, both with the standard CR structure as the unit sphere in $\mathbb{C}^2$ and with a perturbed CR structure known as the Rossi sphere. We show that quotient manifolds from these actions are indeed CR manifolds, and relate the order of the subgroup of $\operatorname{SU}(2)$ to the asymptotic distribution of the Kohn Laplacian's eigenvalues on the quotient. We show that the order of the subgroup determines whether the quotient of the Rossi sphere by the action of that subgroup is CR embeddable. Finally, in the unperturbed case, we prove that we can determine the size of the subgroup by using the point spectrum.

math.CV↗
arXiv2021

Factorization length distribution for affine semigroups IV: a geometric approach to weighted factorization lengths in three-generator numerical semigroups

For numerical semigroups with three generators, we study the asymptotic behavior of weighted factorization lengths, that is, linear functionals of the coefficients in the factorizations of semigroup elements. This work generalizes many previous results, provides more natural and intuitive proofs, and yields a completely explicit error bound.

math.CO↗
arXiv2020

The Cyclic Graph of a Z-group

For a group $G$, we define a graph $\Delta(G)$ by letting $G^{\#} = G \setminus \{ 1 \}$ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\#}$ if and only if the subgroup $\langle x,y\rangle$ is cyclic. Recall that a $Z$-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta(G)$ for a $Z$-group $G$.

math.GR↗
arXiv2020

The cyclic graph (deleted enhanced power graph) of a direct product

Let $G$ be a finite group. Define a graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to be adjacent if and only if $\langle x,y\rangle$ is cyclic. Denote this graph by $Δ(G)$. The graph $Δ(G)$ has appeared in the literature under the names cyclic graph and deleted enhanced power graph. If $G$ and $H$ are nontrivial groups, then $Δ(G\times H)$ is completely characterized. In particular, if $Δ(G\times H)$ is connected, then a diameter bound is obtained, along with an example meeting this bound. Also, necessary and sufficient conditions for the disconnectedness of $Δ(G\times H)$ are established.

math.GR↗
arXiv2019

Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function

Garcia, Kahoro, and Luca showed that the Bateman-Horn conjecture implies $ϕ(p-1) \geq ϕ(p+1)$ for a majority of twin-primes pairs $p,p+2$ and that the reverse inequality holds for a small positive proportion of the twin primes. That is, $p$ tends to have more primitive roots than does $p+2$. We prove that Dickson's conjecture, which is much weaker than Bateman-Horn, implies that the quotients $\frac{ϕ(p+1)}{ϕ(p-1)}$, as $p,p+2$ range over the twin primes, are dense in the positive reals. We also establish several Schinzel-type theorems, some of them unconditional, about the behavior of $\frac{ϕ(p+1)}{ϕ(p)}$ and $\frac{σ(p+1)}{σ(p)}$, in which $σ$ denotes the sum-of-divisors function.

math.NT↗