More Variations on Shuffle Squares

IF 2.2 3区 综合性期刊 Q2 MULTIDISCIPLINARY SCIENCES Symmetry-Basel Pub Date : 2023-10-26 DOI:10.3390/sym15111982
Jarosław Grytczuk, Bartłomiej Pawlik, Mariusz Pleszczyński
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Abstract

We study an abstract variant of squares (and shuffle squares) defined by a constraint graph G, specifying which pairs of words form a square. So, a shuffle G-square is a word that can be split into two disjoint subwords U and W (of the same length), which are joined by an edge. This setting generalizes a recently introduced model of shuffle squares based on word symmetry and permutations. By using the probabilistic method, we provide a sufficient condition for a constraint graph G guaranteeing the avoidability of shuffle G-squares. By a more-elementary method (known as Rosenfeld counting), we prove that G-squares are avoidable over an alphabet of size 4α, α>1, provided that the degree of every word of length n in G is at most αn. We also introduce the concept of the cutting distance between words and state several conjectures involving this notion and various kinds of shuffle squares. We suspect that, for every k⩾2, there is a constant ck such that every even word can be turned into a shuffle square by cutting it in at most ck places and rearranging the resulting pieces. We present some computational, as well as theoretical evidence in favor of this conjecture.
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洗牌方块的更多变化
我们研究了由约束图G定义的正方形(和洗牌正方形)的抽象变体,指定哪些单词对构成正方形。因此,洗牌g平方是一个可以分成两个不相交的子词U和W(长度相同)的词,它们由一条边连接。这种设置概括了最近引入的基于单词对称和排列的洗牌方块模型。利用概率方法,给出了约束图G保证洗牌G-squares可避免性的充分条件。通过一种更基本的方法(称为Rosenfeld计数),我们证明了在一个大小为4α, α>1的字母表上,如果G中每个长度为n的单词的度数最多为αn,则G平方是可以避免的。我们还介绍了单词之间切割距离的概念,并提出了几个关于这个概念和各种洗牌方块的猜想。我们怀疑,对于每个k或2,有一个恒定的ck使得每个偶数单词可以通过在最多ck处切割并重新排列结果片段而变成洗牌方块。我们提出了一些支持这一猜想的计算证据和理论证据。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Symmetry-Basel
Symmetry-Basel MULTIDISCIPLINARY SCIENCES-
CiteScore
5.40
自引率
11.10%
发文量
2276
审稿时长
14.88 days
期刊介绍: Symmetry (ISSN 2073-8994), an international and interdisciplinary scientific journal, publishes reviews, regular research papers and short notes. Our aim is to encourage scientists to publish their experimental and theoretical research in as much detail as possible. There is no restriction on the length of the papers. Full experimental and/or methodical details must be provided, so that results can be reproduced.
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