The set of hook lengths of an integer partition λ is the complement of some numerical semigroup S. There has been recent interest in studying the number of partitions with a given set of hook lengths. Very little is known about the distribution of sizes of this finite set of partitions. We focus on the problem of determining the size of the smallest partition with its set of hook lengths equal to ℕ∖ S.

Recently, there has been significant interest in applying the method of moments developed by Wood and others to study distributions of finite abelian groups that arise in number theory and combinatorics. When the moments do not grow too fast, they determine a unique distribution. We construct large families of distributions that have the same moments. These families include several distributions that arise naturally in the study of sandpile groups of families of random graphs. Wood determined the distribution of Sylow p-subgroups of sandpile groups of Erdős–Rényi random graphs. This was extended by Mészáros to sandpile groups of random d-regular graphs, who observed an interesting special case when d is even and p = 2. We study Sylow p-subgroups of sandpile groups of random bipartite graphs and similarly find a special case for p =2. Although this distribution differs from that of Mészáros, we show that they have the same moments and fit into our broader construction. To compute the moments of the distributions we study, we apply combinatorial tools from the theory of Hall–Littlewood functions.

We study statistical properties of random numerical semigroups of a given genus. We analyze the graph of a typical numerical semigroup, understood as a function from ℕ to ℕ. If S is a numerical semigroup of genus g, this leads us to consider the collection of points (k-1/g-1,a_k(S)/g) where 1 ≤ k ≤ g and a_k(S) denotes the kth smallest nonzero element of S. We show that as g →∞, this set of points typically becomes closer to a union of two line segments. We prove analogous results for numerical semigroups ordered by Frobenius number.

In this paper we investigate the p-rank stratification of the moduli space of curves of genus g that admit a double cover to a fixed elliptic curve E in characteristic p>2. We show that the closed p-rank strata of this moduli space are equidimensional of the expected dimension. We also show the existence of a smooth double cover of E of all the possible values of the p-rank on this moduli space.

Consider a finite field $\mathbb{F}_q$ and positive integers $d,m,r$ with $1\leq r\leq \binom{m+d}{d}$. Let $S_d(m)$ be the $\mathbb{F}_q$ vector space of all homogeneous polynomials of degree $d$ in $X_0,\dots,X_m$. Let $e_r(d,m)$ be the maximum number of $\mathbb{F}_q$-rational points in the vanishing set of $W$ as $W$ varies through all subspaces of $S_d(m)$ of dimension $r$. Ghorpade, Datta and Beelen had conjectured an exact formula of $e_r(d,m)$ when $q\geq d+1$. We prove that their conjectured formula is true when $q$ is sufficiently large in terms of $m,d,r$. The problem of determining $e_r(d,m)$ is equivalent to the problem of computing the $r^{th}$ generalized hamming weights of projective the Reed Muller code $PRM_q(d,m)$. It is also equivalent to the problem of determining the maximum number of points on sections of Veronese varieties by linear subvarieties of codimension $r$.

Given a family of odd abelian covers of ℙ^1 and a prime p of good reduction, in [1], under some assumptions, we computed the generic Newton polygon (resp. Ekedahl-Oort type) in the family (called p-ordinary). In this paper, we investigate the existence of non-p-ordinary smooth curves in the family. In particular, under some restrictions, we show that when p is sufficiently large, the complement of the p-ordinary locus is always nonempty. For 1-dimensional families satisfying additional auxiliary conditions, we obtain a lower bound for the number of non-p-ordinary smooth curves. In specific instances, the above general statement can be improved; as an example, for families of covers of degree at most 7, we establish the non-emptiness of certain non-p-ordinary Newton/Ekedahl-Oort strata (called almost p-ordinary). Our method relies on further study of the extended Hasse-Witt matrix introduced by Moonen in [2], and initiated in [1], and on known results about the geometry of the mod-p reduction of Shimura varieties of PEL type.

Given a family of abelian covers of $\mathbb{P}^1$ and a prime $p$ of good reduction, by considering the associated Deligne--Mostow Shimura variety, we obtain lower bounds for the Ekedahl-Oort type, and the Newton polygon, at $p$ of the curves in the family. In this paper, we investigate whether such lower bounds are sharp. In particular, we prove sharpness when the number of branching points is at most five and $p$ sufficiently large. Our result is a generalization under stricter assumptions of [2, Theorem 6.1] by Bouw, which proves the analogous statement for the $p$-rank, and it relies on the notion of Hasse-Witt triple introduced by Moonen in [9].

Denote the set of algebraic numbers as [Formula: see text] and the set of algebraic integers as [Formula: see text]. For [Formula: see text], consider its irreducible polynomial in [Formula: see text], [Formula: see text]. Denote [Formula: see text]. Drungilas, Dubickas and Jankauskas show in a recent paper that [Formula: see text]. Given a number field [Formula: see text] and [Formula: see text], we show that there is a subset [Formula: see text], for which [Formula: see text]. We prove that [Formula: see text] is a principal ideal domain if and only if the primes in [Formula: see text] generate the class group of [Formula: see text]. We show that given [Formula: see text], we can find a finite set [Formula: see text], such that for every number field [Formula: see text], we have [Formula: see text]. We study how this set [Formula: see text] relates to the ring [Formula: see text] and the ideal [Formula: see text] of [Formula: see text]. We also show that [Formula: see text] satisfy [Formula: see text] if and only if [Formula: see text] for all number fields [Formula: see text].

We fix a number field [Formula: see text] and study statistical properties of the ring [Formula: see text] as [Formula: see text] varies over algebraic numbers of a fixed degree [Formula: see text]. Given [Formula: see text], we explicitly compute the density of [Formula: see text] for which [Formula: see text] and show that this does not depend on the number field [Formula: see text]. In particular, we show that the density of [Formula: see text] for which [Formula: see text] is [Formula: see text]. In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define [Formula: see text] to be a certain finite subset of [Formula: see text] and show that [Formula: see text] determines the ring [Formula: see text]. We show that if [Formula: see text] satisfy [Formula: see text], then the events [Formula: see text] and [Formula: see text] are independent. As [Formula: see text], we study the asymptotics of the density of [Formula: see text] for which [Formula: see text].

A numerical set T is a subset of N 0 that contains 0 and has finite complement. The atom monoid of T is the set of x ∈ N 0 such that x + T ⊆ T. Marzuola and Miller introduced the anti-atom problem: how many numerical sets have a given atom monoid? This is equivalent to asking for the number of integer partitions with a given set of hook lengths. We introduce the void poset of a numerical semigroup S and show that numerical sets with atom monoid S are in bijection with certain order ideals of this poset. We use this characterization to answer the anti-atom problem when S has small type.

We study statistical properties of numerical semigroups of genus $g$ as $g$ goes to infinity. More specifically, we answer a question of Eliahou by showing that as $g$ goes to infinity, the proportion of numerical semigroups of genus $g$ with embedding dimension close to $g/\sqrt{5}$ approaches $1$. We prove similar results for the type and weight of a numerical semigroup of genus $g$.

We fix a number field K and study statistical properties of the ring 𝒪_K[γ]∩ K as γ varies over algebraic numbers of a fixed degree n≥ 2. Given k≥ 1, we explicitly compute the density of γ for which 𝒪_K[γ]∩ K =𝒪_K[1/k] and show that this does not depend on the number field K. In particular, we show that the density of γ for which 𝒪_K[γ]∩ K=𝒪_K is ζ(n+1)/ζ(n). In a recent paper the authors defined X(K,γ) to be a certain finite subset of Spec(𝒪_K) and showed that X(K,γ) determines the ring 𝒪_K[γ]∩ K. We show that if 𝔭_1,𝔭_2∈Spec(𝒪_K) satisfy 𝔭_1∩ℤ≠𝔭_2∩ℤ, then the events 𝔭_1∈ X(K,γ) and 𝔭_2∈ X(K,γ) are independent. As t→∞, we study the asymptotics of the density of γ for which |X(K,γ)|=t.

We fix a number field $K$ and study statistical properties of the ring $\mathcal{O}_K[\gamma]\cap K$ as $\gamma$ varies over algebraic numbers of a fixed degree $n\geq 2$. Given $k\geq 1$, we explicitly compute the density of $\gamma$ for which $\mathcal{O}_K[\gamma]\cap K =\mathcal{O}_K[1/k]$ and show that this does not depend on the number field $K$. In particular, we show that the density of $\gamma$ for which $\mathcal{O}_K[\gamma]\cap K=\mathcal{O}_K$ is $\frac{\zeta(n+1)}{\zeta(n)}$. In a recent paper the authors defined $X(K,\gamma)$ to be a certain finite subset of $\text{Spec}(\mathcal{O}_K)$ and showed that $X(K,\gamma)$ determines the ring $\mathcal{O}_K[\gamma]\cap K$. We show that if $\mathfrak{p}_1,\mathfrak{p}_2\in \text{Spec}(\mathcal{O}_K)$ satisfy $\mathfrak{p}_1\cap \mathbb{Z}\neq\mathfrak{p}_2\cap \mathbb{Z}$, then the events $\mathfrak{p}_1\in X(K,\gamma)$ and $\mathfrak{p}_2\in X(K,\gamma)$ are independent. As $t\to\infty$, we study the asymptotics of the density of $\gamma$ for which $|X(K,\gamma)|=t$.

A generalized numerical semigroup is a submonoid $S$ of $\mathbb{N}^d$ for which the complement $\mathbb{N}^d\setminus S$ is finite. The points in the complement $\mathbb{N}^d\setminus S$ are called gaps. A gap $F$ is considered Frobenius allowable if there is some relaxed monomial ordering on $\mathbb{N}^d$ with respect to which $F$ is the largest gap. We characterize the Frobenius allowable gaps of a generalized numerical semigroup. A generalized numerical semigroup that has only one maximal gap under the natural partial ordering of $\mathbb{N}^d$ is called a Frobenius generalized numerical semigroup. We show that Frobenius generalized numerical semigroups are precisely those whose Frobenius gap does not depend on the relaxed monomial ordering. We estimate the number of Frobenius generalized numerical semigroup with a given Frobenius gap $F=(F^{(1)},\dots,F^{(d)})\in\mathbb{N}^d$ and show that it is close to $\sqrt{3}^{(F^{(1)}+1)\cdots (F^{(d)}+1)}$ for large $d$. We define notions of quasi-irreducibility and quasi-symmetry for generalized numerical semigroups. While in the case of $d=1$ these notions coincide with irreducibility and symmetry, they are distinct in higher dimensions.

A numerical semigroup is a sub-monoid of the natural numbers under addition that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number f and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number f have genus close to $$\frac{3f}{4}$$ . We denote the number of numerical semigroups of Frobenius number f by N(f). While N(f) is not monotonic we prove that $$N(f)<N(f+2)$$ for every f.

Denote the set of algebraic numbers as ℚ and the set of algebraic integers as ℤ. For γ∈ℚ, consider its irreducible polynomial in ℤ[x], F_γ(x)=a_nx^n+…+a_0. Denote e(γ)=(a_n,a_n-1,…,a_1). Drungilas, Dubickas and Jankauskas show in a recent paper that ℤ[γ]∩ℚ={α∈ℚ|{p| v_p(α)<0}⊆{p| p|e(γ)}}. Given a number field K and γ∈ℚ, we show that there is a subset X(K,γ)⊆Spec(𝒪_K), for which 𝒪_K[γ]∩ K={α∈ K|{𝔭| v_𝔭(α)<0}⊆ X(K,γ)}. We prove that 𝒪_K[γ]∩ K is a principal ideal domain if and only if the primes in X(K,γ) generate the class group of 𝒪_K. We show that given γ∈ℚ, we can find a finite set S⊆ℤ, such that for every number field K, we have X(K,γ)={𝔭∈Spec(𝒪_K)|𝔭∩ S≠∅}. We study how this set S relates to the ring ℤ[γ] and the ideal 𝔇_γ={a∈ℤ| aγ∈ℤ} of ℤ. We also show that γ_1,γ_2∈ℚ satisfy 𝔇_γ_1=𝔇_γ_2 if and only if X(K,γ_1)=X(K,γ_2) for all number fields K.

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].

A numerical set with Frobenius number f is a subset T of N that contains zero and max(N/T) = f. Each numerical set has an associated semigroup A(T) = {t vertical bar t + T subset of T}, which has the same Frobenius number as T. For a fixed Frobenius number f there are 2(f-1) numerical sets. It is known that there is a number gamma close to 0.484 such that the ratio of these numerical sets that are mapped to N-f = {0} boolean OR (f, infinity) is asymptotically gamma. We identify a collection of families N(D, f) of numerical semigroups such that for a fixed D the ratio of the 2(f-1) numerical sets that are mapped to N(D, f) converges to a positive limit as f goes to infinity. We denote the limit as gamma(D), these constants sum up to 1 meaning that they asymptotically account for almost all numerical sets.

A numerical set $S$ is a cofinite subset of $\mathbb{N}$ which contains $0$. We use the natural bijection between numerical sets and Young diagrams to define a numerical set $\widetilde{S}$, such that their Young diagrams are complements. We determine various properties of $\widetilde{S}$, particularly with an eye to closure under addition (for both $S$ and $\widetilde{S}$), which promotes a numerical set to become a numerical semigroup.