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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: BOTTOM SCHUR FUNCTIONS. | BOTTOM SCHUR FUNCTIONS Peter Clifford CNRI Dublin Institute of Technology Ireland peterc@alum.mit.edu Richard P. Stanley 1 Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139 USA rstan@math.mit.edu Submitted Nov 19 2003 Accepted Aug 27 2004 Published Sep 24 2004 MR Subject Classifications 05E05 05E10 Abstract We give a basis for the space spanned by the sum Sa of the lowest degree terms in the expansion of the Schur symmetric functions Sa in terms of the power sum symmetric functions Pp where deg pi 1. These lowest degree terms correspond to minimal border strip tableaux of A. The dimension of the space spanned by Sa where A is a partition of n is equal to the number of partitions of n into parts differing by at least 2. Applying the Rogers-Ramanujan identity the generating function also counts the number of partitions of n into parts 5k 1 and 5k 1. We also show that a symmetric function closely related to sa has the same coefficients when expanded in terms of power sums or augmented monomial symmetric functions. 1 Introduction Let A A1 A2 . be a partition of the integer n i.e. A1 A2 0 and M Ai n. The length A of a partition A is the number of nonzero parts of A. The Durfee or Frobenius rank of A denoted rank A is the length of the main diagonal of the diagram of A or equivalently the largest integer i for which Ai i. The rank of A is the least integer r such that A is a disjoint union of r border strips defined below . Nazarov and Tarasov 1 Sect. 1 in connection with tensor products of Yangian modules defined a generalization of rank to skew partitions or skew diagrams A p. The paper 3 Proposition 2.2 gives several simple equivalent definitions of rank A p . One of the definitions is that rank A p is the least integer r such that A p is a disjoint union of r border strips. It develops a general theory of minimal border strip tableaux of skew shapes introducing the concepts of the snake sequence and the interval set of a skew shape A
