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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 $Δ$ and its Stanley-Reisner ideal $I_Δ$, which arises as the edge ideal of the complement graph of $Δ$, we describe the shape of the Betti table of $I_Δ^{[p]}$ and we give a combinatorial characterization of when $I_Δ^{[p]}$ has a linear resolution.

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.

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.

Seasonal forecasting remains challenging due to the inherent chaotic nature of atmospheric dynamics. This paper introduces DeepSeasons, a novel deep learning approach designed to enhance the accuracy and reliability of seasonal forecasts. Leveraging advanced neural network architectures and extensive historical climatic datasets, DeepSeasons identifies complex, nonlinear patterns and dependencies in climate variables with similar or improved skill respcet GCM-based forecasting methods, at a significant lower cost. The framework also allow tailored application to specific regions or variables, rather than the overall problem of predicting the entire atmosphere/ocean system. The proposed methods also allow for direct predictions of anomalies and time-means, opening a new approach to long-term forecasting and highlighting its potential for operational deployment in climate-sensitive sectors. This innovative methodology promises substantial improvements in managing climate-related risks and decision-making processes.

Particle production in hadronic collisions can be studied in the low-momentum (soft) and high-momentum (hard) transfer regimes. While the latter can be well understood with in perturbative QCD the former contains non-perturbative effects which cannot be calculated from first principles. There is also an intermediate regime called semihard, in which the momentum transfer runs typically from $~1$ to $~10$ GeV. As the hadron-hadron collision energy increases, we expect to see a relative growth of the number of semihard events. It has been conjectured that this growth would be the cause of some changes observed in the multiplicity distributions measured in proton - proton collisions. In this note we revisit the separation between soft and semihard events using the formalism of $k_T$ factorization. The separation is implemented through the introduction of a scale that is the cutoff $Λ$ in the transverse momentum of the produced gluon and allows us to compute the average number of particles produced in each regime. These numbers are used as input in the double negative binomial fit of data, from which we can extract correlations between the fraction of semihard events and the violation o

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-Poincaré 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.

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.

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.

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.

In this paper we investigate Cohen-Macaulayness, Gorensteinness and the Hilbert-Poincaré 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.

In this work we estimate the $N_{ψ(2S)} / N_{J/ψ}$ yield ratio in heavy-ion collisions, considering the interactions of the $ψ(2S) $ and $J/ψ$ states with light mesons in the hadron gas formed at the late stages of these collisions. Starting from the appropriate effective Lagrangians, we first compute the thermally-averaged cross sections for the production and absorption of the mentioned states, and then use them as input in the rate equations to determine the time evolution of $N_{ψ(2S)}$, $N_{J/ψ}$ and $N_{ψ(2S)} / N_{J/ψ}$. The main conclusion of our study is that the $ψ(2S) $ and $J/ψ$ multiplicities do not change much in the hadron gas phase and that the $ψ(2S)$ is more absorbed than the $J/ψ$. The obtained final ratio is in qualitative agreement with experimental data.

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.

In this paper we provide a description of the package \textit{PolyominoIdeals} for \textit{Macaulay2} that allows to deal with collections of cells, polyominoes and related binomial ideals.

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.

Very recently, the two-photon decay width of the $η_b$ meson was computed with lattice QCD methods. This decay has not yet been measured. The knowledge of this width allows for the calculation of the $η_b$ production cross section through photon-photon interactions in ultra-peripheral $PbPb$ collisions. In this work we present this calculation, which is the first application of the lattice result. Since UPCs are gaining an increasing attention of the heavy ion community, we take the opportunity to perform a comprehensive study of the different ways of defining ultra-peripheral collisions and of the different ways to treat the equivalent photon flux.

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.

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.

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.

In this work we study charmonium production in high multiplicity proton-proton collisions. We investigate the role of the spatial distribution of partons in the protons and assume that the proton has a Y shape. In this configuration quarks are more at the surface and gluons in the inner part of the proton. Going from peripheral to more central and then to ultra-central proton-proton collisions, we go from quark-quark collisions to gluon-gluon collisions. Since gluons are much more abundant, the cross sections grow. In the case of charm production this growth is enhanced by the fact that, $σ( g + g \to c + \bar{c}) >> σ( q + \bar{q} \to c + \bar{c})$. These effects can explain the growth seen in the data.