有理$$\varvec{q}，\varvec{t}$$-Catalan多项式的一个猜想公式

IF 0.6 4区数学 Q4 MATHEMATICS, APPLIED Annals of Combinatorics Pub Date : 2023-09-07 DOI:10.1007/s00026-023-00662-2

Graham Hawkes

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引用次数: 0

摘要

我们猜想了有理 q，t-卡塔兰多项式 ${\mathcal {C}}_{r/s}$ 的公式，根据定义，它在 q 和 t 中是对称的。这个猜想认为 ${\mathcal {C}}_{r/s}$ 可以用最大戴克路径索引的对称单项式串来写。我们证明，对于任何有限的（d^*\），给出我们关于无限函数集 \(\{{mathcal {C}_{r/s}^d: r\equiv 1 \mod s, \,\,\, d \le d^*\}) 的猜想的组合证明等价于一个有限计数问题。

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A Conjectured Formula for the Rational $\varvec{q},\varvec{t}$-Catalan Polynomial

We conjecture a formula for the rational q, t-Catalan polynomial ${\mathcal {C}}_{r/s}$ that is symmetric in q and t by definition. The conjecture posits that ${\mathcal {C}}_{r/s}$ can be written in terms of symmetric monomial strings indexed by maximal Dyck paths. We show that for any finite $d^*$, giving a combinatorial proof of our conjecture on the infinite set of functions $\{ {\mathcal {C}}_{r/s}^d: r\equiv 1 \mod s, \,\,\, d \le d^*\}$ is equivalent to a finite counting problem.

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来源期刊

Annals of Combinatorics 数学-应用数学

CiteScore

1.00

自引率

0.00%

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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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