广义Delta猜想的Schröder情形
The Schröder case of the generalized Delta conjecture
摘要：
我们证明了Haglund，Remmel和Wilson（Haglund et al. 2018）关于$Δ_{h_m}Δ_{e_{n-k-1}}'e_n$的猜想的Schröder情况，即$\langle \cdot,e_{n-d}h_d \rangle$情况，我们称之为\emph{generalized Delta conjecture}。这个结果扩展了在（D 'Adderio，Vanden Wyngaerd 2017）中证明的Delta猜想的Schröder情况，这反过来又推广了Haglund的$q,t$ -Schröder（Haglund 2004）。证明给出了这些多项式的递归，扩展了上述特殊情况下已知的多项式。此外，我们给出了另一个组合解释相同的多项式在一个新的反弹统计。此外，我们给出了同一多项式在双修饰的双多角形方面的两个更多的解释，扩展了（D 'Adderio，Iraci 2017）中的一些结果，这反过来又扩展了（Aval et al. 2014）中的结果。此外，我们提供组合双射解释这些解释之间的一些等价。
英文摘要：
We prove the Schröder case, i.e. the case $\langle \cdot,e_{n-d}h_d \rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for $Δ_{h_m}Δ_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This result extends the Schröder case of the Delta conjecture proved in (D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the $q,t$-Schröder of Haglund (Haglund 2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another combinatorial interpretation of the same polynomial in terms of a new bounce statistic. Moreover, we give two more interpretations of the same polynomial in terms of doubly decorated parallelogram polyominoes, extending some of the results in (D'Adderio, Iraci 2017), which in turn extended results in (Aval et al. 2014). Also, we provide combinatorial bijections explaining some of the equivalences among these interpretations.
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。
