多维数组的二项式复杂度
Binomial Complexity of Multidimensional Arrays
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中文总结 AI 辅助
本文研究多维数组的二项式复杂度，通过Magnus变换计算列二项式系数，定义\\((k,\ell)\\)-二项式等价与复杂度函数，并给出二维Thue--Morse数组的精确公式。
中文摘要 AI 辅助
多维数组的二项式系数用于统计特定配置的出现次数。为便于阐述，本文重点讨论二维有限数组的列二项式系数。首先，我们证明这些系数可以通过某种Magnus变换来计算。为了获取数组的结构和组合信息，我们随后定义了有限数组的\\((k,\ell)\\)-二项式等价关系，并由此定义了无限数组的\\((k,\ell)\\)-二项式复杂度函数。粗略地说，两个有限数组在\\((k,\ell)\\)-二项式上等价，当且仅当它们包含的尺寸至多为\\(k\times \ell\\)的子数组数量相同。我们获得了编码无限词直积的无限数组的\\((k,\ell)\\)-二项式复杂度的一般结果。我们的主要定理给出了二维Thue--Morse数组的\\((k,\ell)\\)-二项式复杂度的精确公式。为此，我们仔细考察了按位补码对Thue--Morse词的因子在\\(k\\)-二项式等价类上的作用。
英文摘要
Binomial coefficients for multidimensional arrays count occurrences of particular configurations. For the sake of presentation, the emphasis is put on column-binomial coefficients for two-dimensional finite arrays. In this article, we first show that these coefficients can be computed through some Magnus transform. To get structural and combinatorial information on arrays, we then define \((k,\ell)\)-binomial equivalence for finite arrays and, from it, the \((k,\ell)\)-binomial complexity function of an infinite array. Roughly, two finite arrays are \((k,\ell)\)-binomially equivalent when they share the same number of subarrays of size at most $k\times \ell$. We obtain general results on the \((k,\ell)\)-binomial complexity of infinite arrays coding direct products of infinite words. Our main theorem gives an exact formula for the \((k,\ell)\)-binomial complexity of the two-dimensional Thue--Morse array. To that end, we closely examine the action of the bit-wise complement on the \(k\)-binomial equivalence classes of the factors of the Thue--Morse word.
发表机构
- Univ. of Liège（列日大学）
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