论文标题
具有算术约束的排列
Permutations with arithmetic constraints
论文作者
论文摘要
令$ s _ {\ rm lcm}（n）$表示$ [n] = \ {n] = \ {1,2，\ dots，n \} $的排列集$ {\ rm lcm} [j，rm lcm} [j，j，j，j）]此外，令$ s _ {\ rm div}（n）$表示$ [n] $的排列$π$π$，这样$ j \midπ（j）$或$π（j）\ mid j $ in [n] $中的每个$ j \。显然，$ s _ {\ rm div}（n）\ subset s _ {\ rm lcm}（n）$。我们获得了这些集合计数的上限和下限，表明它们几何生长。我们还从最近的论文中证明了关于$ [n] $的“反省”排列数量的猜想，这意味着每个$ \ gcd（j，j，j））> 1 $，除非$ j = 1 $。
Let $S_{\rm lcm}(n)$ denote the set of permutations $π$ of $[n]=\{1,2,\dots,n\}$ such that ${\rm lcm}[j,π(j)]\le n$ for each $j\in[n]$. Further, let $S_{\rm div}(n)$ denote the number of permutations $π$ of $[n]$ such that $j\midπ(j)$ or $π(j)\mid j$ for each $j\in[n]$. Clearly $S_{\rm div}(n)\subset S_{\rm lcm}(n)$. We get upper and lower bounds for the counts of these sets, showing they grow geometrically. We also prove a conjecture from a recent paper on the number of "anti-coprime" permutations of $[n]$, meaning that each $\gcd(j,π(j))>1$ except when $j=1$.
