Baxter permutations originally arose in studying common fixed points of two commuting continuous functions. In 2015, Dilks proposed a conjectured bijection between Baxter permutations and non-intersecting triples of lattice paths in terms of inverse descent bottoms, descent positions and inverse descent tops. We prove this bijectivity conjecture by investigating its connection with the Françon–Viennot bijection. As a result, we obtain a permutation interpretation of the $(t,q)$-analog of the Baxter numbers$\frac{1}{{\left[\begin{array}{c}n+1\\ 1\end{array}\right]}_q{\left[\begin{array}{c}n+1\\ 2\end{array}\right]}_q}\sum _{k=0}^{n-1}{q}^{3\left(\begin{array}{c}k+1\\ 2\end{array}\right)}{\left[\begin{array}{c}n+1\\ k\end{array}\right]}_q{\left[\begin{array}{c}n+1\\ k+1\end{array}\right]}_q{\left[\begin{array}{c}n+1\\ k+2\end{array}\right]}_q{t}^k,$ where ${\left[\begin{array}{c}n\\ k\end{array}\right]}_q$ denote the q-binomial coefficients.

中文翻译：

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Dilks 关于 Baxter 排列的双射猜想的证明

巴克斯特排列最初是在研究两个可交换连续函数的公共不动点时出现的。2015年，Dilks在逆下降底部、下降位置和逆下降顶部方面提出了巴克斯特排列和不相交格路径三元组之间的猜想双射。我们通过研究它与弗朗松-维诺双射的关系来证明这个双射猜想。结果，我们得到了一个排列解释$（t,q）$-巴克斯特数的模拟$\frac{1}{{\left[\begin{array}{c}n+1\\ 1\end{array}\right]}_q{\left[\begin{array}{c}n+1\\ 2\end{array}\right]}_q}\underset{k=0}{\overset{n-1}{\Sigma }}{q}^{3（\begin{array}{c}k+1\\ 2\end{array}）}{\left[\begin{array}{c}n+1\\ k\end{array}\right]}_q{\left[\begin{array}{c}n+1\\ k+1\end{array}\right]}_q{\left[\begin{array}{c}n+1\\ k+2\end{array}\right]}_q{t}^k,$在哪里${\left[\begin{array}{c}n\\ k\end{array}\right]}_q$表示q二项式系数。