Geodesic
Canonical characters on quasi-symmetric functions and bivariate Catalan numbers
The electronic journal of combinatorics, The Stanley Festschrift volume, Tome 11 (2004) no. 2
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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.
DOI : 10.37236/1872
Classification : 05E05, 05A15, 16W50
Mots-clés : Hopf algebra, character, central binomial coefficient, peak of a permutation 
@article{10_37236_1872,
     author = {Marcelo Aguiar and Samuel K. Hsiao},
     title = {Canonical characters on quasi-symmetric functions and bivariate {Catalan} numbers},
     journal = {The electronic journal of combinatorics},
     year = {2004},
     volume = {11},
     number = {2},
     doi = {10.37236/1872},
     zbl = {1071.05072},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1872/}
}
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Marcelo Aguiar; Samuel K. Hsiao. Canonical characters on quasi-symmetric functions and bivariate Catalan numbers. The electronic journal of combinatorics, The Stanley Festschrift volume, Tome 11 (2004) no. 2. doi: 10.37236/1872

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