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Counting and signed counting permutations by descent-based statistics

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Abstract

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 formulas exist in the literature, including formulas of Carlitz and Scoville (Discrete Math 5:45–59, 1973; J Reine Angew Math 265:110–137, 1974), J. Combin. Theory Ser. A, 20: 336-356 (1976), Zhuang (Adv Appl Math 90:86–144, 2017), Pan and Zeng (Adv Appl Math 104:85–99, 2019; Discrete Math 346:113575, 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, which provide two generalizations of Désarménien and Foata’s classical signed Eulerian identity.

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Data sharing was not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. This run network was first used in [42] to count permutations by peaks.

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Acknowledgements

We wish to thank Yan Zhuang whose helpful suggestions lead to the results in Sect. 3.2, Yeong-Nan Yeh for his useful discussions on context-free grammars, Shaoshi Chen for verifying (3.14) using Zeilberger’s algorithm and Qiongqiong Pan for providing a generating function proof of (3.14). This work was supported by the National Science Foundation of China Grants 12271301 and 12322115.

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  1. Research Center for Mathematics and Interdisciplinary Sciences, Shandong University and Frontiers Science Center for Nonlinear Expectations, Ministry of Education, Qingdao, 266237, People’s Republic of China

    Yao Dong & Zhicong Lin

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  1. Yao Dong
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Dong, Y., Lin, Z. Counting and signed counting permutations by descent-based statistics. J Algebr Comb (2024). https://doi.org/10.1007/s10801-024-01330-1

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