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Keywords:
Hopf algebra; simple graph; quasi-symmetric function; character
Hopf algebra; simple graph; quasi-symmetric function; character
Summary:
A multiplicative functional on a graded connected Hopf algebra is called the character. Every character decomposes uniquely as a product of an even character and an odd character. We apply the character theory of combinatorial Hopf algebras to the Hopf algebra of simple graphs. We derive explicit formulas for the canonical characters on simple graphs in terms of coefficients of the chromatic symmetric function of a graph and of canonical characters on quasi-symmetric functions. These formulas and properties of characters are used to derive some interesting numerical identities relating multinomial and central binomial coefficients.
References:
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[2] Aguiar, M., Hsiao, S. K.: Canonical characters on quasi-symmetric functions and bivariate Catalan numbers. Electron. J. Comb. 11 (2005), Research paper R15 34 pp. MR 2120110 | Zbl 1071.05072
[3] Schmitt, W. R.: Incidence Hopf algebras. J. Pure Appl. Algebra 96 (1994), 299-330. DOI 10.1016/0022-4049(94)90105-8 | MR 1303288 | Zbl 0808.05101
[4] Stanley, R.: A symmetric function generalization of the chromatic polynomial of a graph. Adv. Math. 111 (1995), 166-194. DOI 10.1006/aima.1995.1020 | MR 1317387 | Zbl 0831.05027
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