关于排列上两类特殊下降的两个猜想
On two conjectures concerning special kinds of descents on permutations
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中文总结 AI 辅助
本文证明了Han等人关于排列的2型下降数与循环数的生成函数的连分数猜想，建立了对称群上双统计量(des₂, ear)的对称性并给出其等分布伴随统计量，解决了Baril等人相关猜想的重新表述问题。
中文摘要 AI 辅助
本文中，我们证明了Han、Mao和Zeng提出的、关于排列的生成函数（按2型下降数和循环数计数）的连分数猜想，从而解决了他们对Baril和Kirgizov最初提出的一个猜想的重新表述。我们进一步证明了Han、Mao和Zeng猜想的、对称群𝔖ₙ上双统计量(des₂, ear)的对称性，并通过展示(des₂, ear)的五个等分布伴随统计量来强化该结果。其中，统计量des₂表示2型下降的数量，统计量ear表示Sokal和Zeng最初引入的独占反记录循环峰的数量。
英文摘要
In this paper, we prove the continued fraction conjecture posed by Han, Mao and Zeng for the generating function of permutations with respect to the number of descents of type $2$ and the number of cycles, thereby settling their reformulation of a conjecture originally due to Baril and Kirgizov. We further establish the symmetry of the bistatistic $(\des_2, \ear)$ over $\mathfrak{S}_n$ as conjectured by Han, Mao and Zeng and strengthen this result by exhibiting five equidistributed companions for $(\des_2, \ear)$. Here the statistic $\des_2$ denotes the number of descents of type $2$, and the statistic $\ear$ denotes the number of exclusive antirecord cycle peaks originally introduced by Sokal and Zeng.
