  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.
