Geodesic
A note on three types of quasisymmetric functions
The electronic journal of combinatorics, Tome 12 (2005)

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In the context of generating functions for $P$-partitions, we revisit three flavors of quasisymmetric functions: Gessel's quasisymmetric functions, Chow's type B quasisymmetric functions, and Poirier's signed quasisymmetric functions. In each case we use the inner coproduct to give a combinatorial description (counting pairs of permutations) to the multiplication in: Solomon's type A descent algebra, Solomon's type B descent algebra, and the Mantaci-Reutenauer algebra, respectively. The presentation is brief and elementary, our main results coming as consequences of $P$-partition theorems already in the literature.
DOI : 10.37236/1958
Classification : 05E05, 16S34
Mots-clés : Gessel quasisymmetric functions, Chow quasisymmetric functions, Poirier signed quasisymmetric functions, Solomon descent algebra, Mantaci-Reutenauer algebra 
T. Kyle Petersen. A note on three types of quasisymmetric functions. The electronic journal of combinatorics, Tome 12 (2005). doi: 10.37236/1958
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     author = {T. Kyle Petersen},
     title = {A note on three types of quasisymmetric functions},
     journal = {The electronic journal of combinatorics},
     year = {2005},
     volume = {12},
     doi = {10.37236/1958},
     zbl = {1088.05075},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1958/}
}
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