$\lambda$-Factorials of $n$

Abstract

Recently, by the Riordan identity related to tree enumerations, \begin{align*} \sum_{k=0}^{n}\binom{n}{k}(k+1)!(n+1)^{n-k} = (n+1)^{n+1}, \end{align*} Sun and Xu have derived another analogous one, \begin{align*} \sum_{k=0}^{n}\binom{n}{k}D_{k+1}(n+1)^{n-k} = n^{n+1}, \end{align*} where $D_{k}$ is the number of permutations with no fixed points on $\{1,2,\dots, k\}$. In the paper, we utilize the $\lambda$-factorials of $n$, defined by Eriksen, Freij and W$\ddot{a}$stlund, to give a unified generalization of these two identities. We provide for it a combinatorial proof by the functional digraph theory and two algebraic proofs. Using the umbral representation of our generalized identity and Abel's binomial formula, we deduce several properties for $\lambda$-factorials of $n$ and establish interesting relations between the generating functions of general and exponential types for any sequence of numbers or polynomials.