In the 1995 paper entitled “Noncommutative symmetric functions”, Gelfand et al. defined two noncommutative symmetric function analogues for the power sum basis of the symmetric functions. This paper explores the combinatorial properties of their duals, two distinct quasisymmetric power sum bases. In contrast to the symmetric power sums, the quasisymmetric power sums have a more complex combinatorial description. This paper offers a first detailed exploration of these two relatively unstudied quasisymmetric bases, in which we show that they refine the classical symmetric power sum basis, we give transition matrices to other well-understood bases, and we provide explicit formulas for products of quasisymmetric power sums.

在1995年题为“非对换对称函数”的论文中，Gelfand等人。为对称函数的幂和定义了两个非交换对称函数类似物。本文探讨了它们的对偶，两个不同的拟对称幂和基的组合性质。与对称幂和相反，准对称幂和具有更复杂的组合描述。本文首先对这两个相对未被研究的拟对称基进行了详细的探讨，其中我们证明了它们完善了经典的对称幂和基，我们将过渡矩阵赋予了其他易于理解的基，并为拟对称功率的乘积提供了明确的公式总和。