Skip to main content

Refined Restricted Inversion Sequences


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  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].


  1. 1.

    D. André, Développement de \(\sec \,x\) and tg \(x\), C. R. Math. Acad. Sci. Paris, 88 (1879), 965–979.

  2. 2.

    J.-L. Baril and V. Vajnovszki, A permutation code preserving a double Eulerian bistatistic, Discrete Appl. Math., 224 (2017), 9–15.

  3. 3.

    M. Barnabei, F. Bonetti, M. Silimbani, The descent statistic on \(123\)-avoiding permutations, Sém. Lothar. Combin., 63 (2010), Art. B63a.

  4. 4.

    M. Bousquet-Mélou, Four classes of pattern-avoiding permutations under one roof: generating trees with two labels, Electron. J. Combin., 9 (2003), #R19.

  5. 5.

    C.-O. Chow, W.C. Shiu, Counting simsun permutations by descents, Ann. Comb., 15 (2011), 625–635.

  6. 6.

    F.R.K. Chung, R.L. Graham, V.E. Hoggatt, Jr. and M. Kleiman, The number of Baxter permutations, J. Combin. Theory Ser. A, 24 (1978), 382–394.

  7. 7.

    S. Connolly, Z. Gabor and A. Godbole, The location of the first ascent in a 123-avoiding permutation, Integers, 15 (2015), #A13.

  8. 8.

    S. Corteel, M. Martinez, C.D. Savage and M. Weselcouch, Patterns in Inversion Sequences I, Discrete Math. Theor. Comput. Sci., 18 (2016), \(\#2\).

  9. 9.

    D. Dumont, Interprétations combinatoires des numbers de Genocchi (in French), Duke Math. J., 41 (1974), 305–318.

  10. 10.

    S. Elizalde, Fixed points and excedances in restricted permutations, Electron. J. Combin., 18(2) (2012), #P29.

  11. 11.

    D. Foata, Distributions eulériennes et mahoniennes sur le groupe des permutations, in M. Aigner (ed.), Higher combinatorics, pp. 27–49, Boston, Dordrecht, 1977.

  12. 12.

    D. Foata and G.-N. Han, André Permutation Calculus: a twin Seidel matrix sequence, Sém. Lothar. Combin., 73 (2014), Art. B73e, 54 pp.

  13. 13.

    D. Foata and M.-P. Schützenberger, Nombres d’Euler et permutations alternantes, in A Survey of Combinatorial Theory, J.N. Srivistava, et al., eds., North-Holland, Amsterdam, 1973, pp. 173–187.

  14. 14.

    S. Fu, Z. Lin and J. Zeng, On two new unimodal descent polynomials, Discrete Math., 341 (2018), 2616–2626.

  15. 15.

    G. Hetyei, On the \(cd\)-variation polynomials of André and Simsun permutations, Discrete Comput. Geom., 16 (1996), 259–275.

  16. 16.

    S. Kitaev, Patterns in permutations and words, Springer Science & Business Media, 2011.

  17. 17.

    C. Krattenthaler, Permutations with restricted patterns and Dyck paths, Special issue in honor of Dominique Foata’s 65th birthday, Adv. Appl. Math., 27 (2001), 510–530.

  18. 18.

    D. Kremer, Permutations with forbidden subsequences and a generalized Schröder numbers, Discrete Math., 218 (2000), 121–130.

  19. 19.

    Z. Lin and D. Kim, A sextuple equidistribution arising in Pattern Avoidance, J. Combin. Theory Ser. A, 155 (2018), 267–286.

  20. 20.

    T. Mansour, E.Y.P. Deng and R.R.X. Du, Dyck paths and restricted permutations, Discrete Appl. Math., 154 (2006), 1593–1605.

  21. 21.

    T. Mansour and M. Shattuck, Pattern avoidance in inversion sequences, Pure Math. Appl. (PU.M.A.), 25 (2015), 157–176.

  22. 22.

    T. Mansour, personal communication, March 2021.

  23. 23.

    M.A. Martinez and C.D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, J. Integer Seq., 21 (2018), Article 18.2.2. (arXiv:1609.08106v1).

  24. 24.

    C. Poupard, De nouvelles significations énumératives des nombres d’Entringer, Discrete Math., 38 (1982), 265–271.

  25. 25.

    C.D. Savage and M. Visontai, The s-Eulerian polynomials have only real roots, Trans. Amer. Math. Soc., 367 (2015), 1441–1466.

  26. 26.

    R. Stanley, Enumerative combinatorics. Volume 1. Second edition. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, Cambridge, 2012.

  27. 27.

    S. Sundaram, The homology of partitions with an even number of blocks, J. Algebraic Combin., 4 (1995), 69–92.

Download references


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).

Author information



Corresponding author

Correspondence to Dongsu Kim.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Éric Fusy.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lin, Z., Kim, D. Refined Restricted Inversion Sequences. Ann. Comb. (2021).

Download citation


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