Abstract We construct a non-commutative, non-cocommutative, graded bialgebra Π with a basis indexed by the permutations in all finite symmetric groups. Unlike the formally similar Malvenuto–Poirier–Reutenauer Hopf algebra, this bialgebra does not have finite graded dimension. After giving formulas for the product and coproduct, we show that there is a natural morphism from Π to the algebra of quasi-symmetric functions, under which the image of a permutation is its associated Stanley symmetric function. As an application, we use this morphism to derive some new enumerative identities. We also describe analogues of Π for the other classical types. In these cases, the relevant objects are module coalgebras rather than bialgebras, but there are again natural morphisms to the quasi-symmetric functions, under which the image of a signed permutation is the corresponding Stanley symmetric function of type B, C, or D.

中文翻译：

---

Stanley 对称函数的双代数

摘要 我们构造了一个非交换、非交换、分级双代数 Π，其基由所有有限对称群中的置换索引。与形式上相似的 Malvenuto-Poirier-Reutenauer Hopf 代数不同，这种双代数没有有限的分级维数。在给出乘积和余积的公式之后，我们证明了从 Π 到拟对称函数的代数存在一个自然态射，在这种情况下，置换的图像是其相关的斯坦利对称函数。作为一个应用，我们使用这个态射来推导出一些新的枚举恒等式。我们还描述了其他经典类型的 Π 的类似物。在这些情况下，相关对象是模余代数而不是双代数，但对于拟对称函数也有自然态射，