基于上下文无关文法的三角数组
Triangular Arrays using context-free grammar
AI总结：
本文利用上下文无关文法与组合微分方程的关系，将三角数组T(n,k)解释为递增树，并推导出显式公式和结构性质，特别涉及r-Whitney-Eulerian数和特定参数下的结果。
摘要：
本文工作利用Hao文法G={u→u^{b1+b2+1}v^{a1+a2}, v→u^{b2}v^{a2+1}}，结合文法与组合微分方程的对应关系，以获得形式为T(n,k)=(a2n+a1k+a0)T(n−1,k)+(b2n+b1k+b0)T(n−1,k−1)的三角数组的解释。这导致T(n,k)被解释为递增树。通过分析微分方程推导出显式公式和结构性质。特别是，得到了r-Whitney-Eulerian数以及当b2n+b1k+b0=1的情况的显式表达式。应用包括r-Eulerian数的新解释公式及其生成函数。我们还利用此方法得到了a2=−a1情况下的完整生成函数。
英文摘要：
In this work, the Hao grammar $G=\{\, u\rightarrow u^{b_1+b_2+1} v^{a_1+a_2},\quad v\rightarrow u^{b_2}v^{a_2+1} \,\},$ together with the correspondence between grammars and combinatorial differential equations, is employed to obtain an interpretation of any triangular array of the form \[ T(n,k)=(a_2 n + a_1 k + a_0)\,T(n-1,k) + (b_2 n + b_1 k + b_0)\,T(n-1,k-1). \] This lead to have an interpretation of $T(n,k)$ as an increasing tree. Explicit formulas and structural properties are then derived through analytic differential equations. In particular, the $r$-Whitney-Eulerian numbers and the cases where $b_2n+b_1k+b_0=1$ are obtained explicitly. \noindent Applications include new interpretation formulas for the $r$-Eulerian numbers with generating functions. We also obtain full generating functions for the case $a_2=-a_1$ using this approach.
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。
