Dumont 的一个文法与 Diaconis-Evans-Graham 定理
A Grammar of Dumont and a Theorem of Diaconis-Evans-Graham
AI总结：
本文发现 Dumont 文法与 Diaconis-Evans-Graham 定理的关联，利用文法演算推导生成函数，给出左成功数类比和文法辅助双射，并证明相关等分布。
摘要：
我们发现了 Dumont 关于 $S_n$ 上 $(\exc, \fix)$ 联合分布的一个非凡文法与 Diaconis-Evans-Graham 关于排列的成功数和不动点的优美定理之间的意外联系。借助该文法，我们展示了文法演算在推导生成函数方面的优势，其中常数性质起着重要作用。基于排列的左成功数，我们对 Roselle 研究的联合分布给出了一种文法处理。此外，我们获得了 Diaconis-Evans-Graham 定理的左成功数类比，体现了文法辅助双射的思想。文法标记引出了 $(\u200b\jump, \des)$ 与 $(\exc, \drop)$ 在左成功数集合和不动点集合上的等分布，其中 $\jump$ 定义为上升数减去左成功数。
英文摘要：
We came across an unexpected connection between a remarkable grammar of Dumont for the joint distribution of $(\exc, \fix)$ over $S_n$ and a beautiful theorem of Diaconis-Evans-Graham on successions and fixed points of permutations. With the grammar in hand, we demonstrate the advantage of the grammatical calculus in deriving the generating functions, where the constant property plays a substantial role. On the grounds of left successions of a permutation, we present a grammatical treatment of the joint distribution investigated by Roselle. Moreover, we obtain a left succession analogue of the Diaconis-Evans-Graham theorem, exemplifying the idea of a grammar assisted bijection. The grammatical labelings give rise to an equidistribution of $(\jump, \des)$ and $(\exc, \drop)$ restricted to the set of left successions and the set of fixed points, {where $\jump$ is defined to be the number of ascents minus the number of left successions.}
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
