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

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$

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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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