报告题目：The Hecke Insertion Algorithm and Two Combinatorial Problems

报告人：陈永川 院士 天津大学应用数学中心

时间：5月10号 16:00

地点：管研楼1518

摘要：The Hecke insertion algorithm was developed by Buch, Kresch, Shimozono,Tamvakis and Yong [Math. Ann., 2008] in order to expand a stable Grothendieckpolynomial in terms of stable Grothendieck polynomials indexed by integer parti-tions. The notion of a rook strip was introduced by Buch [Acta Math., 2002] inthe study of the Littlewood-Richardson rule for stable Grothendieck polynomials.We introduce the notion of Hecke diagrams and the structure of vacillating Hecketableaux. A Hecke diagram is defined as a Young diagram possibly with a markedcorner. A vacillating Hecke tableaux is a sequence of Hecke diagrams subject tocertain conditions on the addition and deletion of rook strips. Using the Heckeinsertion algorithm, we establish a one-to-one correspondence between vacillatingHecke tableaux and linked partitions which arise in free probability theory. Basedon this correspondence, we show that the crossing number and the nesting numberhave a symmetric joint distribution over linked partitions, confirming a conjec-ture of de Mier [Electron. J. Combin., 2006]. We also prove a conjecture of Kim[SIAM J. Discrete Math., 2011] which states that the crossing number and thenesting number have a symmetric joint distribution over the front representationsof partitions.

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