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Noncommutative Catalan Numbers

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Abstract

The goal of this paper is to introduce and study noncommutative Catalan numbers\(C_n\) which belong to the free Laurent polynomial algebra \(\mathcal {L}_n\) in n generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia–Haiman (q, t)-versions, another—to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices \(H_n\) and introduce accompanying noncommutative binomial coefficients.

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References

  1. Aigner, M.: Catalan-like numbers and determinants. J. Combin. Theory Ser. A 87(1), 33–51 (1999)

    Article  MathSciNet  Google Scholar 

  2. Andrews, G.: Catalan numbers, \(q\)-Catalan numbers and hypergeometric series. J. Combin. Theory Ser. A 44(2), 267–273 (1987)

    Article  MathSciNet  Google Scholar 

  3. Berenstein, A., Retakh, V.: A short proof of Kontsevich’s cluster conjecture. C. R. Math. Acad. Sci. Paris 349(3-4), 119–122 (2011)

    Article  MathSciNet  Google Scholar 

  4. Berenstein, A., Retakh, V.: Noncommutative marked surfaces. Adv. Math. 328, 1010–1087 (2018)

    Article  MathSciNet  Google Scholar 

  5. Berenstein, A., Retakh, V., Reutenauer, C., Zeilberger, D.: The reciprocal of \(\sum _{n\ge 0}a^nb^n\) for non-commuting \(a\) and \(b\), Catalan numbers and non-commutative quadratic equations. In: Berenstein, A., Retakh, V. (eds.) Noncommutative birational geometry, representations and combinatorics, Contemp. Math., 592, pp. 103–109. Amer. Math. Soc., Providence, RI (2013)

  6. Carlitz, L.: Sequences, paths, ballot numbers. Fibonacci Quart. 10(5), 531–549 (1972)

    MathSciNet  MATH  Google Scholar 

  7. Carlitz, L., Riordan, J.: Two element lattice permutation numbers and their \(q\)-generalization. Duke Math. J. 31, 371–388 (1964)

    Article  MathSciNet  Google Scholar 

  8. Di Francesco, Ph., Kedem, R.: Non-commutative integrability, paths and quasi-determinants. Adv. Math. 228(1), 97–152 (2011)

    Article  MathSciNet  Google Scholar 

  9. Fürlinger, J., Hofbauer, J.: \(q\)-Catalan numbers. J. Combin. Theory Ser. A 40(2), 248–264 (1985)

    Article  MathSciNet  Google Scholar 

  10. Garsia, A.M., Haiman, M.: A remarkable \(q,t\)-Catalan sequence and \(q\)-Lagrange inversion. J. Algebraic Combin. 5(3), 191–244 (1996)

    Article  MathSciNet  Google Scholar 

  11. Garsia, A.M., Haglund, J.: A proof of the \(q,t\)-Catalan positivity conjecture. Discrete Math. 256(3), 677–717 (2002)

    Article  MathSciNet  Google Scholar 

  12. Gelfand, I., Gelfand, S., Retakh, V., Wilson, R.L.: Quasideterminants. Adv. Math. 193(1), 56–141 (2005)

    Article  MathSciNet  Google Scholar 

  13. Gelfand, I.M., Krob, D., Lascoux, A., Leclerc, B., Retakh, V.S., Thibon, J.-Y.: Noncommutative symmetric functions. Adv. Math. 112(2), 218–348 (1995)

    Article  MathSciNet  Google Scholar 

  14. Gelfand, I.M., Retakh, V.S.: Theory of noncommutative determinants, and characteristic functions of graphs. Funct. Anal. Appl. 26(4), 231–246 (1992)

    Article  MathSciNet  Google Scholar 

  15. Gelfand, I.M., Retakh, V.S.: Determinants of matrices over noncommutative rings. Funct. Anal. Appl. 25(2), 91–102 (1991)

    Article  MathSciNet  Google Scholar 

  16. Haglund, J.: The \(q,t\)-Catalan Numbers and the Space of Diagonal Harmonics. With an appendix on the combinatorics of Macdonald polynomials. University Lecture Series, 41. Amer. Math. Soc. Providence, RI (2008)

  17. Pak, I., Postnikov, A., Retakh, V.: Noncommutative Lagrange theorem and inversion polynomials. (preprint) https://math.mit.edu/~apost/papers/noncom.pdf

  18. Stanley, R.P.: Catalan Numbers. Cambridge University Press, New York (2015)

    Book  Google Scholar 

  19. Tamm, U.: Some aspects of Hankel matrices in coding theory and combinatorics. Electron. J. Combin. 8(1), Art. 1, 31 pp (2001)

Download references

Acknowledgements

This work was partly done during our visits to Max-Planck-Institut für Mathematik and Institut des Hautes Études Scientifiques. We gratefully acknowledge the support of these institutions. We thank Philippe Di Francesco and Rinat Kedem for their comments on the first version of the paper, particularly for explaining to us a relationship between noncommutative Stieltjes continued fractions and our noncommutative Catalan series (see Remark 2.6).

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Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403, USA

    Arkady Berenstein

  2. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Vladimir Retakh

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  1. Arkady Berenstein
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  2. Vladimir Retakh
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Correspondence to Vladimir Retakh.

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Dedicated to Professor George Andrews on the occasion of his eightieth birthday.

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This work was partially supported by the NSF Grant DMS-1403527 (A. B.)

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Berenstein, A., Retakh, V. Noncommutative Catalan Numbers. Ann. Comb. 23, 527–547 (2019). https://doi.org/10.1007/s00026-019-00473-4

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