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Refined Restricted Inversion Sequences

Abstract

Recently, the study of patterns in inversion sequences was initiated by Corteel–Martinez–Savage–Weselcouch and Mansour–Shattuck independently. Motivated by their works and a double Eulerian equidistribution due to Foata (1977), we investigate several classical statistics on restricted inversion sequences that are either known or conjectured to be enumerated by Catalan, Large Schröder, Baxter and Euler numbers. One of the two highlights of our results is a fascinating bijection between 000-avoiding inversion sequences and Simsun permutations, which together with Foata’s V- and S-codes, provide a proof of a restricted double Eulerian equidistribution. The other one is a refinement of a conjecture due to Martinez and Savage that the cardinality of \({\mathbf{I}}_n(\ge ,\ge ,>)\) is the n-th Baxter number, which is proved via the so-called obstinate kernel method developed by Bousquet-Mélou.

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Notes

  1. 1.

    This bijection is essentially due to Krattenthaler [17]. An alternative description of \(\psi \) using zigzag strips was presented in [20].

  2. 2.

    This conjecture has been proved recently by Mansour and Shattuck [22].

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Acknowledgements

We thank Éric Fusy for his insightful comments and suggestions on improving this paper and one anonymous referee for drawing our attention to [20]. This work was supported by the National Science Foundation of China grants 11871247 and 11501244, by the Austrian Science Foundation FWF, START grant Y463 and SFB grant F50, by the project of Qilu Young Scholars of Shandong University, and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2019R1F1A1062462).

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Correspondence to Dongsu Kim.

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Communicated by Éric Fusy.

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Lin, Z., Kim, D. Refined Restricted Inversion Sequences. Ann. Comb. (2021). https://doi.org/10.1007/s00026-021-00550-7

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Keywords

  • Inversion sequences
  • Ascents
  • Distinct entries
  • Last entry
  • Schröder numbers
  • Baxter numbers