Seminars on Discrete Mathematics: Counting and signed counting permutations by descent-based statistics

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:2024-04-09 10:00





The original motivation of this paper was to find the context-free grammar for the joint distribution of peaks and valleys on permutations. Although such attempt was unsuccessful, we can obtain noncommutative symmetric function identities for the joint distributions of several descent-based statistics, including peaks, valleys and even/odd descents, on permutations via Zhuang's generalized run theorem. Our results extend in a unified way several generating function formulae exist in the literature, including formulas of Carlitz--Scoville (1973, 1974), Stanley (1976), Zhuang (2017) and Pan--Zeng (2019, 2023). As applications of these generating function formulas, Wachs' involution and Foata--Strehl action on permutations, we also investigate the signed counting of even and odd descents, and of descents and peaks.

This talk is based on joint work with Yao Dong. 


林志聪,山东大学数学与交叉科学研究中心教授,国家优青获得者。主要从事计数组合学的研究,在《J. Combin. Theory Ser. A》、《Combinatorica》、《European J. Combin.》、《Proc. Amer. Math. Soc.》等权威期刊发表SCI学术论文40余篇。任中国数学会计算机数学专业委员会委员和中国运筹学会图论组合分会青年理事。