\(\lambda \)-factorials of \(n\). (English) Zbl 1204.05008
Summary: Recently, by the Riordan identity related to tree enumerations,
\[ \sum_{k=0}^n \binom nk (k+1)!(n+1)^{n-k}= (n+1)^{n+1}, \]
Sun and Xu have derived another analogous one,
\[ \sum_{k=0}^n \binom nk D_{k+1}(n+1)^{n-k}= n^{n+1}, \]
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 \(n\), denned by Eriksen, Freij and Wästlund, 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.
\[ \sum_{k=0}^n \binom nk (k+1)!(n+1)^{n-k}= (n+1)^{n+1}, \]
Sun and Xu have derived another analogous one,
\[ \sum_{k=0}^n \binom nk D_{k+1}(n+1)^{n-k}= n^{n+1}, \]
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 \(n\), denned by Eriksen, Freij and Wästlund, 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.
MSC:
| 05A05 | Permutations, words, matrices |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 05A40 | Umbral calculus |
| 05C05 | Trees |
