山谷三角洲猜想的一些结果

IF 0.6 4区 数学 Q4 MATHEMATICS, APPLIED
Michele D’Adderio, Alessandro Iraci
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引用次数: 0

摘要

Haglund et al. (Trans Am Math Soc 370(6): 4029-4057, 2018)介绍了他们的Delta猜想,该猜想给出了对称函数$$\Delta '_{e_{n-k-1}} e_n$$ Δ en - k - 1 ' en的两种不同的组合解释,以上升装饰或山谷装饰的标记Dyck路径。虽然上升版本最近已被证明(D 'Adderio and Mellit in Adv Math 402:108342, 2022;Blasiak等人在《扩展Delta猜想的证明》(A Proof of Extended Delta Conjecture, arXiv:2102.08815, 2021)中指出,对于山谷版本的了解并不多。在这项工作中,我们证明了谷三角洲猜想的Schröder情况,其正方形版本的Schröder情况(Iraci和Wyngaerd In Ann Combin 25(1):195 - 227,2021),以及其扩展版本的加泰罗尼亚情况(Qiu和Wilson In J Combin Theory Ser A 175:105271, 2020)。此外,假设扩展山谷三角洲猜想的组合侧的对称性(一种改进),我们还推导出其方形版本的加泰罗尼亚情况(Iraci和Wyngaerd 2021)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Some Consequences of the Valley Delta Conjectures

Some Consequences of the Valley Delta Conjectures
Abstract Haglund et al. (Trans Am Math Soc 370(6):4029–4057, 2018) introduced their Delta conjectures , which give two different combinatorial interpretations of the symmetric function $$\Delta '_{e_{n-k-1}} e_n$$ Δ e n - k - 1 e n in terms of rise-decorated or valley-decorated labelled Dyck paths. While the rise version has been recently proved (D’Adderio and Mellit in Adv Math 402:108342, 2022; Blasiak et al. in A Proof of the Extended Delta Conjecture, arXiv:2102.08815 , 2021), not much is known about the valley version. In this work, we prove the Schröder case of the valley Delta conjecture, the Schröder case of its square version (Iraci and Wyngaerd in Ann Combin 25(1):195–227, 2021), and the Catalan case of its extended version (Qiu and Wilson in J Combin Theory Ser A 175:105271, 2020). Furthermore, assuming the symmetry of (a refinement of) the combinatorial side of the extended valley Delta conjecture, we deduce also the Catalan case of its square version (Iraci and Wyngaerd 2021).
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来源期刊
Annals of Combinatorics
Annals of Combinatorics 数学-应用数学
CiteScore
1.00
自引率
0.00%
发文量
56
审稿时长
>12 weeks
期刊介绍: Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board. The scope of Annals of Combinatorics is covered by the following three tracks: Algebraic Combinatorics: Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices Analytic and Algorithmic Combinatorics: Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms Graphs and Matroids: Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches
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