$\lambda$-factorials of $n$

TL;DR

This paper introduces a unified generalization of two combinatorial identities involving factorials and derangements using $mbda$-factorials, supported by combinatorial and algebraic proofs, and explores their generating functions.

Contribution

It provides the first unified generalization of these identities using $mbda$-factorials, along with multiple proofs and properties of the generalized sequence.

Findings

Derived a generalized identity involving $mbda$-factorials.

Proved the identity using combinatorial and algebraic methods.

Explored relations between generating functions of the sequence.

Abstract

Recently, by the Riordan's identity related to tree enumerations, \begin{eqnarray*} \sum_{k=0}^{n}\binom{n}{k}(k+1)!(n+1)^{n-k} &=& (n+1)^{n+1}, \end{eqnarray*} Sun and Xu derived another analogous one, \begin{eqnarray*} \sum_{k=0}^{n}\binom{n}{k}D_{k+1}(n+1)^{n-k} &=& n^{n+1}, \end{eqnarray*} 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 another two algebraic proofs. Using the umbral representation of our generalized identity and the Abel's binomial formula, we deduce several properties for $\lambda$-factorials of $n$ and establish the curious relations between the generating functions of general and exponential types for any sequence of numbers or polynomials.