{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:23:59Z","timestamp":1759335839889,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>In this paper, we give a decomposition of triply rooted trees into three doubly rooted trees. This leads to a combinatorial interpretation of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory, and proved by Younsi by using the Hurwitz identity on multivariate Abel polynomials. We also give a bijection between the set of functions from [n+1] to [n] and the set of triply rooted trees on [n], which leads to the refined enumeration of functions from [n+1] to [n] with respect to the number of elements in the orbit of n+1 and the number of periodic points.<\/jats:p>","DOI":"10.37236\/3016","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:13:29Z","timestamp":1578719609000},"source":"Crossref","is-referenced-by-count":5,"title":["Decomposition of Triply Rooted Trees"],"prefix":"10.37236","volume":"20","author":[{"given":"William Y.C.","family":"Chen","sequence":"first","affiliation":[]},{"given":"Janet F.F.","family":"Peng","sequence":"additional","affiliation":[]},{"given":"Harold R.L.","family":"Yang","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2013,4,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p10\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v20i2p10\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T11:22:47Z","timestamp":1579260167000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v20i2p10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,4,17]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2013,4,9]]}},"URL":"https:\/\/doi.org\/10.37236\/3016","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2013,4,17]]},"article-number":"P10"}}