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Francesco Navarra

Publications and source records attributed to Francesco Navarra.

At least 19 recordsLinked to original sources
arXiv2026

On the Linearity of Squarefree Powers of Edge Ideals

Let $G$ be a graph and $I(G)$ its edge ideal. The $p$-th squarefree power $I(G)^{[p]}$ is the monomial ideal generated by squarefree monomials corresponding to the matchings of size $p$ of $G$. In this paper, we provide a combinatorial characterization of when $I(G)^{[p]}$ is linearly related, i.e., when its first syzygy module is generated by linear forms. Moreover, for a $1$-dimensional flag simplicial complex $\Delta$ and its Stanley-Reisner ideal $I_{\Delta}$, which arises as the edge ideal of the complement graph of $\Delta$, we describe the shape of the Betti table of $I_{\Delta}^{[p]}$ and we give a combinatorial characterization of when $I_{\Delta}^{[p]}$ has a linear resolution.

math.AC↗
arXiv2026

An algebraic study of ideals of weak graph homomorphisms

Let $G$ and $H$ be finite simple graphs and assume that either both are undirected or both are directed. We introduce and study the ideal of weak graph homomorphisms $I_{G\to H}$. We characterize all graphs $G$ and $H$ for which every (equivalently, some) power of $I_{G\to H}$ has a linear resolution. Moreover, unmixedness, Cohen-Macaulayness, projective dimension and Castelnuovo-Mumford regularity of these ideals are studied.

math.AC↗
arXiv2026

The Complete Intersection property for binomial ideals of collections of cells

In this paper, we provide a combinatorial characterization of those collections of cells whose inner $2$-minor ideals are complete intersections. More precisely, given a collection of cells $\mathcal C$ and its associated inner $2$-minor ideal $I_{\mathcal C}$, we prove that $I_{\mathcal C}$ is a complete intersection if and only if $\mathcal C$ is a chessboard.

math.AC↗
arXiv2025

Minimal primes and radicality of ideals generated by adjacent 2-minors

In this paper, we provide a complete description of the minimal primes of ideals generated by adjacent $2$-minors, in terms of the so-called admissible sets and associated lattice ideals. We prove that for these ideals, the properties of being unmixed, Cohen-Macaulay, level, Gorenstein, and complete intersection are equivalent. Moreover, we give a combinatorial characterization of all convex collections of cells satisfying any of these equivalent properties. Finally, we study the radicality of these ideals and derive necessary combinatorial conditions based on minimal non-radical configurations.

math.AC↗
arXiv2025

Switching rook polynomial of collections of cells

We explore the novel connection between rook placements on collections of cells, also known as pruned chessboards, and the algebraic properties of ideals generated by $2$-minors. We design an algorithm to compute the switching rook polynomial of a collection of cells and show that it coincides with the $h$-polynomial of the associated coordinate ring for all collections up to rank 10 and polyominoes up to rank 12. Motivated by this evidence, we conjecture that the correspondence holds in general, and we prove it for certain convex collections of cells by algebraic tools.

math.AC↗
arXiv2025

Recent Advances in the Theory of Polyomino Ideals

Polyomino ideals, defined as the ideals generated by the inner $2$-minors of a polyomino, are a class of binomial ideals whose algebraic properties are closely related to the combinatorial structure of the underlying polyomino. We provide a unified account of recent advances on two central themes: the characterization of prime polyomino ideals and the emerging connection between the Hilbert-Poincar\'e series and Gorensteinness of $K[\mathcal{P}]$ with the classical rook theory. Some further related properties, as radicality, primary decomposition, and levelness are discussed, and a \textit{Macaulay2} package, namely \texttt{PolyominoIdeals}, is also presented.

math.AC↗
arXiv2025

$t$-Young complexes and squarefree powers of $t$-path ideals

We introduce a new class of simplicial complexes, called \emph{$t$-Young complexes}, arising from a Young diagram and a positive integer~$t$. We show that every $t$-Young complex is either contractible or homotopy equivalent to a wedge of spheres. A complete characterization of their vertex-decomposability is provided, and in several cases, we establish explicit formulas for their homotopy types. Interestingly, $t$-Young complexes naturally appear as the Alexander dual complexes of squarefree powers of $t$-path ideals of path graphs, as well as of certain ideals generated by subsets of their minimal generators. As an application, we derive formulas for the projective dimension and Krull dimension of these squarefree powers.

math.AC↗
arXiv2025

Switching Rook Polynomials of Collections of Cells: Palindromicity and Domino-Stability

The rook polynomial is a generating function that enumerates the number of ways to place rooks, with no two in the same row or column, on a collection of cells regarded as a pruned chessboard. In combinatorial commutative algebra, special attention is devoted to its variant, the switching rook polynomial, which is conjectured to coincide with the $h$-polynomial of the $K$-algebra associated with the given collection of cells. In this context, palindromicity plays a crucial role, as it reflects the algebraic property of Gorensteinness. In this paper, we introduce a new combinatorial property, called domino-stability, and we prove that the switching rook polynomial of a collection of cells $\mathcal{P}$ is palindromic if and only if $\mathcal{P}$ is domino-stable. Building upon this result, we derive new insights into the characterization of Gorenstein $K$-algebras arising from polyominoes or, more generally, from collections of cells.

math.CO↗
arXiv2024

Generalized Numerical semigroups up to isomorphism

A generalized numerical semigroup is a submonoid $S$ of $\mathbb{N}^d$ with finite complement in it. We characterize isomorphisms between these monoids in terms of permutation of coordinates. Considering the equivalence relation that identifies the monoids obtained by the action of a permutation and establishing a criterion to select a representative from each equivalence class, we define some procedures for generating the set of all generalized numerical semigroups of given genus up to isomorphism. Finally, we present computational data and explore properties related to the number of generalized numerical semigroups of a given genus up to isomorphism.

math.CO↗
arXiv2024

Shellable flag simplicial complexes of non-simple polyominoes

In this article we investigate the shellability of the flag simplicial complexes attached to non-simple and thin polyominoes. As a consequence, we obtain the Cohen-Macaulayness and a combinatorial interepetation of the $h$-polynomial of the related coordinate rings.

math.AC↗
arXiv2024

On squarefree powers of simplicial trees

In this article, we study the squarefree powers of facet ideals associated with simplicial trees. Specifically, we examine the linearity of their minimal free resolution and their regularity. Additionally, we investigate when the first syzygy module of squarefree powers of a simplicial tree is generated by linear relations. Finally, we provide a combinatorial formula for the regularity of the squarefree powers of $t$-path ideals of path graphs.

math.AC↗
arXiv2024

On Cohen-Macaulay non-prime collections of cells

In this paper we investigate Cohen-Macaulayness, Gorensteinness and the Hilbert-Poincar\'{e} series for some classes of non-prime collections of cells. In particular, we show that all closed path polyominoes are Cohen-Macaulay and we characterize those that are Gorenstein.

math.AC↗
arXiv2023

On the rook polynomial of grid polyominoes

Grid polyominoes form a class of thin polyominoes with one or more holes arranged in a grid-like pattern in the plane. In this paper, we prove that the rook polynomial of grid polyominoes coincides with the h-polynomial of their corresponding coordinate ring. Our approach is based on the theory of simplicial complexes and extends previous results for frame polyominoes, which are special cases of polyominoes with exactly one hole.

math.CO↗
arXiv2023

Shellable simplicial complex and switching rook polynomial of frame polyominoes

Let $\mathcal{P}$ be a frame polyomino, a new kind of non-simple polyomino. In this paper we study the $h$-polynomial of $K[\mathcal{P}]$ in terms of the switching rook polynomial of $\mathcal{P}$ using the shellable simplicial complex $\Delta(\mathcal{P})$ attached to $\mathcal{P}$. We provide a suitable shelling order for $\Delta(\mathcal{P})$ and we define a bijection between the set of the canonical configurations of $j$ rooks in $\mathcal{P}$ and the facets of $\Delta(\mathcal{P})$ with $j$ steps. Finally we use a well-known combinatorial result, due to McMullen and Walkup, about the $h$-vector of a shellable simplicial complex to interpret the $h$-polynomial of $K[\mathcal{P}]$ as the switching rook polynomial of $\mathcal{P}$.

math.CO↗
arXiv2023

Polyocollection ideals and primary decomposition of polyomino ideals

In this article, we study the primary decomposition of some binomial ideals. In particular, we introduce the concept of polyocollection, a combinatorial object that generalizes the definitions of collection of cells and polyomino, that can be used to compute a primary decomposition of non-prime polyomino ideals. Furthermore, we give a description of the minimal primary decomposition of non-prime closed path polyominoes. In particular, for such a class of polyominoes, we characterize the set of all zig-zag walks and show that the minimal prime ideals have a very nice combinatorial description.

math.AC↗
arXiv2023

Hilbert-Poincaré series and Gorenstein property for some non-simple polyominoes

Let $\mathcal{P}$ be a closed path having no zig-zag walks, a kind of non-simple thin polyomino. In this paper we give a combinatorial interpretation of the $h$-polynomial of $K[\mathcal{P}]$, showing that it is the rook polynomial of $\mathcal{P}$. It is known by Rinaldo and Romeo (2021), that if $\mathcal{P}$ is a simple thin polyomino then the $h$-polynomial is equal to the rook polynomial of $\mathcal{P}$ and it is conjectured that this property characterizes all thin polyominoes. Our main demonstrative strategy is to compute the reduced Hilbert-Poincaré series of the coordinate ring attached to a closed path $\mathcal{P}$ having no zig-zag walks, as a combination of the Hilbert-Poincaré series of convenient simple thin polyominoes. As a consequence we prove that the Krull dimension is equal to $\vert V(\mathcal{P})\vert -\mathrm{rank}\, \mathcal{P}$ and the regularity of $K[\mathcal{P}]$ is the rook number of $\mathcal{P}$. Finally we characterize the Gorenstein prime closed paths, proving that $K[\mathcal{P}]$ is Gorenstein if and only if $\mathcal{P}$ consists of maximal blocks of length three.

math.AC↗
arXiv2022

On some classes of generalized numerical semigroups

A generalized numerical semigroup is a submonoid of $\mathbb{N}^d$ with finite complement in it. In this work we study some properties of three different classes of generalized numerical semigroups. In particular, we prove that the first class satisfies a generalization of Wilf's conjecture, by introducing a generalization of a well-known sufficient condition for Wilf's conjecture in numerical semigroups, that involves the type of the semigroup. Partial results for Wilf's generalized conjecture are obtained also for the other two classes, and some open questions are provided.

math.CO↗