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总览 评价 李雪珊 （ ； ） 摘要： 在研究泊车函数上的余交换Hopf代数PFSym时,本文作者曾给出PFSym的两组自由生成元,它们分别由原子泊车函数与不可裂泊车函数标记。原子泊车函数与不可裂泊车函数这两个概念分别是通过斜线号乘积和分裂乘积两个二元运算引入。
李雪珊
（
； ）
摘要：
在研究泊车函数上的余交换Hopf代数PFSym时,本文作者曾给出PFSym的两组自由生成元,它们分别由原子泊车函数与不可裂泊车函数标记。原子泊车函数与不可裂泊车函数这两个概念分别是通过斜线号乘积和分裂乘积两个二元运算引入。由此可知,长度为$n$的原子泊车函数与长度为$n$的不可裂泊车函数个数是相同的。在本文中,我们将在更为一般的自函数框架下,给出该结果的双射证明。我们将斜线号运算与分裂运算推广到自函数上,并提出原子自函数及不可裂自函数的概念。我们证明每个自函数都能唯一分解为原子自函数的斜线号乘积或者分解为不可裂自函数的分裂乘积。此外,我们还证明了,对任意正整数$ngeq 1$, 长度为$n$的原子自函数与长度为$n$的不可裂自函数是一一对应的。
关键词：
自函数；泊车函数；排列；分拆
LI Xueshan
（
； ）
Abstract：
In the study of a cocommutative Hopf algebra PFSym built on parking functions, the author presented two free generating sets of PFSym which are indexed by atomic parking functions and unsplitable parking functions. The notions of atomic parking function and unsplitable parking function are introduced via two binary operations called the slash product and the split product respectively. It follows that the set of atomic parking functions of length $n$ is equinumerous with the set of unsplitable parking functions of length $n$. In this paper, we will give a bijective proof of this fact in the more general setting of endofunctions. We generalize the slash product and the split product to endofunctions, and then propose the notions of atomic endofunction and unsplitable endofunction. We show that each endofunction can be uniquely factorized into slash (resp. split) product of atomic (resp. unsplitable) endofunctions. Moreover, it is shown that for each $ngeq 1$, the set of atomic endofunctions of length $n$ is in bijection with the set of unsplitable endofunctions of length $n$.
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