{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:07Z","timestamp":1753893847011,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.37236\/441","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T03:58:50Z","timestamp":1578715130000},"source":"Crossref","is-referenced-by-count":1,"title":["$\\lambda$-Factorials of $n$"],"prefix":"10.37236","volume":"17","author":[{"given":"Yidong","family":"Sun","sequence":"first","affiliation":[]},{"given":"Jujuan","family":"Zhuang","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,12,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r169\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r169\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:19:10Z","timestamp":1579303150000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r169"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,12,10]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/441","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,12,10]]},"article-number":"R169"}}