%0 Journal Article
%T A note on three types of quasisymmetric functions
%A T. Kyle Petersen
%J Mathematics
%D 2005
%I arXiv
%X 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.
%U http://arxiv.org/abs/math/0508149v1