TAILIEUCHUNG - Báo cáo toán học: "A Bijection between Atomic Partitions and Unsplitable Partitions"

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: A Bijection between Atomic Partitions and Unsplitable Partitions. | A Bijection between Atomic Partitions and Unsplitable Partitions William . Chen1 Teresa . Li2 1 2Center for Combinatorics LPMC-TJKLC Nankai University Tianjin 300071 . China 1chen@ 2lxs@ David . Wang3 3Beijing International Center for Mathematical Research Peking University Beijing 100871 . China 3wgl@ Submitted Sep 3 2010 Accepted Dec 23 2010 Published Jan 5 2011 Mathematics Subject Classification 05A18 05A19 05E05 Abstract In the study of the algebra NCSym of symmetric functions in noncommutative variables Bergeron and Zabrocki found a free generating set consisting of power sum symmetric functions indexed by atomic partitions. On the other hand Bergeron Reutenauer Rosas and Zabrocki studied another free generating set of NCSym consisting of monomial symmetric functions indexed by unsplitable partitions. Can and Sagan raised the question of finding a bijection between atomic partitions and unsplitable partitions. In this paper we provide such a bijection. 1 Introduction In their study of the algebra NCSym of symmetric functions in noncommutative variables Rosas and Sagan 5 introduced a vector space with a basis pn n is a set partition where pn is the power sum symmetric function in noncommutative variables. Bergeron Hohlweg Rosas and Zabrocki 1 obtained the following formula pn ơ pn pơ where n ơ denotes the slash product of n and Ơ. It follows that as an algebra NCSym is freely generated by pn with n atomic see Bergeron and Zabrocki 3 . It should be noted THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P7 1 that Wolf 6 showed that NCSym is freely generated by another basis. A combinatorial characterization of the generating set of Wolf has been found by Bergeron Reutenauer Rosas and Zabrocki 2 . More precisely they introduced the notion of unsplitable partitions and proved that the generating set of Wolf can be described as the set of monomial symmetric functions in noncommutative variables indexed by .

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