论文标题
函数组成和相关组合身份的预期无可逆性程度
The expected degree of noninvertibility of compositions of functions and a related combinatorial identity
论文作者
论文摘要
最近，防御者和PROPP [2020]定义了$ f \ colon x \至y $的非易换性程度。我们获得了$ t $函数$ t $ fuctions $ t \ in \ mathbb {n} $的$ t $ function构图的预期不可逆性的确切公式。那么，对于非可逆性程度的定义，一个等效的公式是概括的起点，从而产生了一种看似新的组合身份，涉及第一类签名的stirling数字的Stirling Transform。
Recently, Defant and Propp [2020] defined the degree of noninvertibility of a function $f\colon X\to Y$ between two finite nonempty sets by $\text{deg}(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. We obtain an exact formula for the expected degree of noninvertibility of the composition of $t$ functions for every $t\in \mathbb{N}$. An equivalent formulation for the definition of the degree of noninvertibility is then the starting point for a generalization yielding a seemingly new combinatorial identity involving the Stirling transform of the signed Stirling numbers of the first kind.
