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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Partitions, rooks, and symmetric functions in noncommuting variables. | Partitions rooks and symmetric functions in noncommuting variables Mahir Bilen Can Department of Mathematics Tulane University New Orleans LA 70118 USA mcan@tulane.edu Bruce E. Sagan Department of Mathematics Michigan State University East Lansing MI 48824-1027 USA sagan@math.msu.edu Submitted Aug 17 2010 Accepted Jan 17 2011 Published Jan 24 2011 Dedicated to Doron Zeilberger on the occasion of his 60th birthday. His enthusiasm for combinatorics has been an inspiration to us all. Key Words noncommuting variables rook set partition symmetric function AMS subject classification 2010 Primary 05A18 Secondary 05E05. Abstract Let nn denote the set of all set partitions of 1 2 . n . We consider two subsets of nn one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let En c nn be the subset of all partitions corresponding to an extendable rook placement on the upper-triangular board Tn-1. Given n E nm and ơ E nn define their slash product to be n ơ nu ơ m E nm n where ơ m is the partition obtained by adding m to every element of every block of ơ. Call T atomic if it can not be written as a nontrivial slash product and let An c nn denote the subset of atomic partitions. Atomic partitions were first defined by Bergeron Hohlweg Rosas and Zabrocki during their study of NCSym the symmetric functions in noncommuting variables. We show that despite their very different definitions En An for all n 0. Furthermore we put an algebra structure on the formal vector space generated by all rook placements on upper triangular boards which makes it isomorphic to NCSym. We end with some remarks. Work partially done while a Program Officer at NSF. The views expressed are not necessarily those of the NsF. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P3 1 1 Extendable rooks and atomic partitions For a nonnegative integer n let n 1 2 . n . Let nn denote the set of all set partitions n of n i.e. n B1 B2 . Bk with oiBi n disjoint union . In .
