论文标题
避免整数排列中的单调算术进程
Avoiding Monotone Arithmetic Progressions in Permutations of Integers
论文作者
论文摘要
整数避免了单调算术算术$ 6 $的置换（Geneson，2018年）。我们通过构建整数的排列来改进这一点，避免了单调算术算术$ 5 $。我们还构建了整数和正整数的置换，以改善先前的上下密度结果。在（Davis等人，1977年）中，他们构建了积极整数的双线无限置换，避免了单调算术算术的长度$ 4 $。我们构建了整数的双线无限置换，以避免单调算术算术$ 5 $。在（Lesaulnier and Vijay，2011年）中构建了避免单调算术算术$ 4 $的单调算术进程的积极整数的排列。我们概括了这个结果，并表明，对于每个$ k \ geq 1 $，存在一个正整数的排列，避免了单调算术算术的长度为$ 4 $的单调算术，而常见的差异不可除以$ 2^k $。此外，我们指定了$ [1，n] $的排列结构，该结构避免了长度$ 3 $单调算术算术进度模式mod $ n $（Davis等，1977），并为避免长度$ k $ k $ k $ monotone arithmetic arithmetic arithmetic arithmetic arithmetic arithmetic arithmetic arithmetic arithmetic arithmetic arithmetic arithmetic arithmetic arithmetic arithmetic arithmetions mod $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $。
A permutation of the integers avoiding monotone arithmetic progressions of length $6$ was constructed in (Geneson, 2018). We improve on this by constructing a permutation of the integers avoiding monotone arithmetic progressions of length $5$. We also construct permutations of the integers and the positive integers that improve on previous upper and lower density results. In (Davis et al. 1977) they constructed a doubly infinite permutation of the positive integers that avoids monotone arithmetic progressions of length $4$. We construct a doubly infinite permutation of the integers avoiding monotone arithmetic progressions of length $5$. A permutation of the positive integers that avoided monotone arithmetic progressions of length $4$ with odd common difference was constructed in (LeSaulnier and Vijay, 2011). We generalise this result and show that for each $k\geq 1$, there exists a permutation of the positive integers that avoids monotone arithmetic progressions of length $4$ with common difference not divisible by $2^k$. In addition, we specify the structure of permutations of $[1,n]$ that avoid length $3$ monotone arithmetic progressions mod $n$ as defined in (Davis et al. 1977) and provide an explicit construction for a multiplicative result on permutations that avoid length $k$ monotone arithmetic progressions mod $n$.
