Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A note on three types of quasisymmetric functions. | A note on three types of quasisymmetric functions T. Kyle Petersen Department of Mathematics Brandeis University Waltham MA USA tkpeters@ Submitted Aug 8 2005 Accepted Nov 14 2005 Published Nov 22 2005 Mathematics Subject Classifications 05E99 16S34 Abstract 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. 1 Quasisymmetric functions and Solomon s descent algebra The ring of quasisymmetric functions is well-known see 12 ch. . Recall that a quasisymmetric function is a formal series Q xi X2 . G Z xi X2 . of bounded degree such that the coefficient of x 1 x x k is the same for all i1 2 k i2 ik and all compositions a a1 a2 . ak . Recall that a composition of n written a n is an ordered tuple of positive integers a a1 a2 . ak such that a a1 a2 ak n. In this case we say that a has k parts or a k. We can put a partial order on the set of all compositions of n by reverse rehnement. The covering relations are of the form oq . ai ơi 1 . ak p ai . ai ai i . ak . Let Qsymn denote the set of all quasisymmetric functions homogeneous of degree n. The ring of quasisymmetric functions can be dehned as Qsym ra 0 Qsymn but our focus will stay on the quasisymmetric functions of degree n rather than the ring as a whole. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R61 1 The most obvious basis for Qsymn is the set of monomial quasisymmetric functions dehned for any composition a 1 a2 . ak 1 n Ma X Xi X2 i1 Ì2 --- ík .
