On some properties of the number of permutations being products of pairwise disjoint $d$-cycles

Permutations and the divisor graph of $[1,n]$

Let $S_{\rm div}(n)$ denote the set of permutations $π$ of $n$ such that for each $1\leq j \leq n$ either $j \mid π(j)$ or $π(j) \mid j$. These permutations can also be viewed as vertex-disjoint directed cycle covers of the divisor graph $\mathcal{D}_{[1,n]}$ on vertices $v_1, \ldots, v_n$ with an edge between $v_i$ and $v_j$ if $i\mid j$ or $j \mid i$. We improve on recent results of Pomerance by showing $c_d = \lim_{n \to \infty }\left(\# S_{\rm div}(n)\right)^{1/n}$ exists and that $2.069<c_d<2.694$. We also obtain similar results for the set $S_{\rm lcm}(n)$ of permutations where ${\rm lcm…

有限域上具有特定圈结构的置换多项式的构造

Construction of Permutation Polynomials of Certain Specific Cycle Structure over Finite Fields

本文针对奇数阶有限域，构造了具有特定圈结构的置换二项式和置换三项式族，包括具有一个不动点且其余元素构成等长互斥圈的置换，以及具有特定频率的多长度圈置换多项式。

关于不相交剩余类的大最大公因数问题

On the problem of large gcd for disjoint residue classes

研究\(k\)个两两不相交剩余类，通过使用边由模的最大公因数染色的完全图等方法，证明了\(\max_{1\leq i<j\leq k}\gcd(m_i,m_j)\)与\(k\)的一个下界关系。

On Sun's conjecture concerning disjoint cosets

In 2004, Zhi-Wei Sun posed the following conjecture: If a_1G_1,...,a_kG_k (k>1) are finitely many pairwise disjoint left cosets in a group G with all the indices [G:G_i] finite, then for some 1\le i<j\le k, the greatest common divisor of [G:G_i] and [G:G_j] is at least k. In this paper, we confirm Sun's conjecture for k=3,4.

费马曲面上的最大斜线数

Maximal number of Skew lines on Fermat Surfaces

本文针对次数$d\geq4$的费马曲面上两两不交直线的最大数量这一未解决问题开展研究，证明了$d$为偶或不等于5的奇数时最大斜线数为$3d$，$d=5$时不超过13条，填补了相关代数几何研究的空白。