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An Analogue of Mahonian Numbers and Log-Concavity

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Abstract

In this paper, we propose a q-analogue of the number of permutations i(n, k) of length n having k inversions known by Mahonian numbers. We investigate useful properties and some combinatorial interpretations by lattice paths/partitions and tilings. Furthermore, we give two constructive proofs of the strong q-log-concavity of the q-Mahonian numbers in k and n, respectively. In particular for \(q=1\), we obtain two constructive proofs of the log-concavity of the Mahonian numbers in k and n, respectively.

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References

  1. Bazeniar, A., Ahmia, M., Belbachir, H.: Connection between bi\(^s\)nomial coefficients with their analogs and symmetric functions. Turkish J. Math. 41(3), 807–818 (2018).

    MATH  Google Scholar 

  2. Bazeniar, A., Ahmia, M., Bouchair, A.: Log-concave sequences of bi\(^{s}\)nomial coefficients with their analogs and symmetric function. Indian J. Pure Appl. Math. 53, 127–137 (2022).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bóna, M.: Combinatorics of Permutations. CRC Press (2004).

  4. Bóna, M.: A Combinatorial Proof of the Log-concavity of a famous sequence counting permutations. Electron. J. Combin. 11(2) (2005), #N2.

  5. Brenti, F.: Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc. 413 (1989).

  6. Butler, L.M.: On the \(q\)-log-concavity of \(q\)-binomial coefficients. J. Combin. Theory Ser. A 54(1), 54–63 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  7. Hardy, G.H., Wright, E. M.: An Introduction to the Theory of Numbers. Oxford, Clarendon Press, (1954).

    MATH  Google Scholar 

  8. Indong, D.J.L., Peralta, G.R.: Inversions of Permutations in Symmetric, Alternating, and Dihedral Groups. J. Integer Seq. 11 (2008).

  9. Knuth, D.E.: Subspaces, subsets, and partitions. J. Combin. Theory Ser. A 10, 178–180 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  10. Knuth, D.E.: The Art of Computer Programming. Addison-Wesley. Reading, MA, Vol. 3, p. 15 (1973).

  11. MacMahon, P.A.: The indices of permutations and the derivation therefrom of functions of a single variable associated with the permutations of any assemblage of objects; Amer. J. Math. 35, 281–322 (1913).

    Article  MathSciNet  MATH  Google Scholar 

  12. Muir: On a simple term of a determinant. Proc. Royal S. Edinborough. 21, 441–477 (1898-9).

  13. Netto, E.: Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, p. 96 (1927).

  14. Sagan, B.E.: Log concave sequences of symmetric functions and analogs of the Jacobi-Trudi determinants. Trans. Amer. Math. Soc. 329(2), 795–811 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  15. Sagan, B.E.: On inductive proofs of \(q\)-log-concavity. Discrete Math. 99, 289–306 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  16. Sloane, N.J.A.: The On-line Encyclopedia of Integer Sequences, https://oeis.org.

  17. Stanley, R.P.: Log-Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry. Annals of the New York Academy of Sciences. 576, 500–553 (1989).

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for many valuable remarks and suggestions to improve the original manuscript. This work was supported by DG-RSDT (Algeria), PRFU Project, No. C00L03UN180120220002.

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  1. LMAM Laboratory, University of Mohamed Seddik Ben Yahia, BP 98 Ouled Aissa, Jijel, 18000, Algeria

    Yousra Ghemit & Moussa Ahmia

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  1. Yousra Ghemit
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  2. Moussa Ahmia
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Correspondence to Moussa Ahmia.

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Communicated by Ken Ono.

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Ghemit, Y., Ahmia, M. An Analogue of Mahonian Numbers and Log-Concavity. Ann. Comb. 27, 895–916 (2023). https://doi.org/10.1007/s00026-022-00614-2

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