1,765 research outputs found

    The number of directed k-convex polyominoes

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    We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed k-convex polyominoes. We show it is a rational function and we study its asymptotic behavior

    A closed formula for the number of convex permutominoes

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    In this paper we determine a closed formula for the number of convex permutominoes of size n. We reach this goal by providing a recursive generation of all convex permutominoes of size n+1 from the objects of size n, according to the ECO method, and then translating this construction into a system of functional equations satisfied by the generating function of convex permutominoes. As a consequence we easily obtain also the enumeration of some classes of convex polyominoes, including stack and directed convex permutominoes

    Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers

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    The first problem addressed by this article is the enumeration of some families of pattern-avoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these being also solved independently by Lin, and Kim and Lin. The strength of our approach is its robustness: we enumerate four families F1⊂F2⊂F3⊂F4F_1 \subset F_2 \subset F_3 \subset F_4 of pattern-avoiding inversion sequences ordered by inclusion using the same approach. More precisely, we provide a generating tree (with associated succession rule) for each family FiF_i which generalizes the one for the family Fi−1F_{i-1}. The second topic of the paper is the enumeration of a fifth family F5F_5 of pattern-avoiding inversion sequences (containing F4F_4). This enumeration is also solved \emph{via} a succession rule, which however does not generalize the one for F4F_4. The associated enumeration sequence, which we call the \emph{powered Catalan numbers}, is quite intriguing, and further investigated. We provide two different succession rules for it, denoted ΩpCat\Omega_{pCat} and Ωsteady\Omega_{steady}, and show that they define two types of families enumerated by powered Catalan numbers. Among such families, we introduce the \emph{steady paths}, which are naturally associated with Ωsteady\Omega_{steady}. They allow us to bridge the gap between the two types of families enumerated by powered Catalan numbers: indeed, we provide a size-preserving bijection between steady paths and valley-marked Dyck paths (which are naturally associated with ΩpCat\Omega_{pCat}). Along the way, we provide several nice connections to families of permutations defined by the avoidance of vincular patterns, and some enumerative conjectures.Comment: V2 includes modifications suggested by referees (in particular, a much shorter Section 3, to account for arXiv:1706.07213

    COMO AS CRIANÇAS APRENDEM E SE DESENVOLVEM: ALGUNS CONHECIMENTOS PARA ENSINAR LÍNGUAS ESTRANGEIRAS A CRIANÇAS

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    Neste artigo descrevemos, brevemente, as principais teorias de aprendizagem e desenvolvimento infantil, um saber que entendemos imprescindível ao professor de espanhol para crianças e, porque não, para todos os professores da Educação Infantil e dos primeiros anos do Ensino Fundamental. Entre as diversas teorias comentadas, apresentamos a maturacionista, que toma como base os postulados de Hall  e Gesell; as vertentes da teoria comportamental de Pavlov, Watson, Thorndike e Skinner; as teorias psicodinâmicas de Freud e Erickson; as ditas psicogenéticas de Piaget e Vygotsky; e a teoria ecológica, de Bronfenbrenner. De conhecimento dessas linhas teóricas, os professores de línguas estrangeiras poderão adequar sua prática ao público infantil com o qual atuam
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