{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:28Z","timestamp":1753893868313,"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>We consider the generating polynomial of the number of rooted trees on the set $\\{1,2,\\dots,n\\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. A classical product formula shows that this polynomial factors completely over the integers.\u00a0From this product formula it can be concluded that this polynomial has positive coefficients in the $\\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of Liu, Dotsenko-Khoroshkin, Bershtein-Dotsenko-Khoroshkin, Gonz\u00e1lez D'Le\u00f3n-Wachs and Gonzl\u00e1ez D'Le\u00f3n related to the free multibracketed Lie algebra and the poset of weighted partitions.<\/jats:p>","DOI":"10.37236\/5361","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T17:02:42Z","timestamp":1578675762000},"source":"Crossref","is-referenced-by-count":1,"title":["A Note on the $\\gamma$-Coefficients of the Tree Eulerian Polynomial"],"prefix":"10.37236","volume":"23","author":[{"given":"Rafael S.","family":"Gonz\u00e1lez D'Le\u00f3n","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,2,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i1p20\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i1p20\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T00:36:38Z","timestamp":1579221398000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i1p20"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2,5]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,1,11]]}},"URL":"https:\/\/doi.org\/10.37236\/5361","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,2,5]]},"article-number":"P1.20"}}