{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:25Z","timestamp":1753893805170,"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>Every character on a graded connected Hopf algebra decomposes uniquely as a product of an even character and an odd character (Aguiar, Bergeron, and Sottile, math.CO\/0310016).  We obtain explicit formulas for the even and odd parts of the universal character on the Hopf algebra of quasi-symmetric functions. They can be described in terms of Legendre's beta function evaluated at half-integers, or in terms of bivariate Catalan numbers: $$C(m,n)={(2m)!(2n)!\\over m!(m+n)!n!}\\,.$$ Properties of characters and of quasi-symmetric functions are then used to derive several interesting identities among bivariate Catalan numbers and in particular among Catalan numbers and central binomial coefficients.<\/jats:p>","DOI":"10.37236\/1872","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:19:09Z","timestamp":1578701949000},"source":"Crossref","is-referenced-by-count":7,"title":["Canonical Characters on Quasi-Symmetric Functions and Bivariate Catalan Numbers"],"prefix":"10.37236","volume":"11","author":[{"given":"Marcelo","family":"Aguiar","sequence":"first","affiliation":[]},{"given":"Samuel K.","family":"Hsiao","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,2,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i2r15\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i2r15\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:51:03Z","timestamp":1579305063000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i2r15"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,2,21]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2004,6,3]]}},"URL":"https:\/\/doi.org\/10.37236\/1872","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2005,2,21]]},"article-number":"R15"}}