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{\bf T. Kyle Petersen}
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{\bf A Note on Three Types of Quasisymmetric Functions}
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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.

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