Faà di Bruno公式在恒等式及计数上的应用

Application of Faà di Bruno’s Formula in Identity and Counting

摘要: Faà di Bruno’ formular的相关研究一直处于停滞状态，之后，由于Faà di Bruno在其它方面的应用，Lukacs开始将Faà di Bruno用于数学统计学；罗曼用umbral calculus的定理对Faà di Bruno方程进行了再论证，Constantine利用Faà di Bruno扩展了关于集合划分的恒等式，并给出了它的概率说明，Chu利用Faà di Bruno获得了一系列的行列式，Chou，Hsu和Shiue利用Faà di Bruno构造了一种具有互逆性的函数，由此推导了一组恒等方程，然而并没有有关用Fáa di Bruno’s公式求组合恒等式以及进行对称群及集合分割上的组合数的研究存在。本文使用Fáa di Bruno’s公式求组合恒等式以及进行对称群及集合分割上的组合数的研究，通过Faà di Bruno得到了各种著名组合数的恒等式，含括Catalan数，第一类及第二类Stirling数，q-二项式系数•••等，及Faà di Bruno在计数上的应用；在第二节中，我们利用Faà di Bruno得到了多种组合数的恒等式；在第三节中，我们由第一类及第二类Stirling数、错排数、Bell数的指数生成函数用Faà di Bruno导出的恒等式获得这些数的组合意义，得到一些限制置换中圈结构和限制集合分割的子集大小的结果。

Abstract: The research on Faà di Bruno has been in a stagnant state. Later, because of the application of Faà di Bruno in other aspects, Lukacs began to use Faà di Bruno in mathematical statistics; Roman demonstrated the Faà di Bruno equation again by using the theorem of umbral calculus. Constan-tine extended the identity of set partition by using Faà di Bruno and gave its probability explana-tion. Chu obtained a series of determinants by using Faà di Bruno, Chou; Hsu and Shiue used Faà di Bruno to construct a kind of reciprocal function, and derived a set of identity equations from it. However, there is no research on finding combinatorial identities by using Fáa di Bruno’s formula and on the number of combinations on symmetric groups and set partitions. Faà di Bruno's formula was used to find the combinatorial identity, and the combinatorial numbers on symmetric groups and set partitions were studied. In this paper, the identities of various famous combinatorial numbers, including Catalan numbers, the first and second Stirling numbers, q binomial coefficients, etc., and the application of Faà di Bruno in counting were obtained through Faà di Bruno; In the second section, we use Faà di Bruno to obtain the identities of multiple combinatorial numbers; in the third section, we obtain the combined meanings of Stirling numbers, staggered numbers and Bell numbers from the exponential generating functions of the first and second types of Stirling numbers with the identities derived from Faà di Bruno, and the subset size of set partitions can be obtained.