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Symmetry and Log-Concavity Results for Statistics on Fibonacci Tableaux

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Abstract

In this paper, we study the properties of the inversion statistic and the Fibonacci major index, Fibmaj, as defined on standard Fibonacci tableaux. We prove that these two statistics are symmetric and log-concave over all standard Fibonacci tableaux of a given shape μ and provide two combinatorial proofs of the symmetry result, one a direct bijection on the set of tableaux and the other utilizing 0, 1-fillings of a staircase shape. We conjecture that the inversion and Fibmaj statistics are log-concave over all standard Fibonacci tableaux of a given size n. In addition, we show a well-known bijection between standard Fibonacci tableaux of size n and involutions in S n which takes the Fibmaj statistic to a new statistic called the submajor index on involutions.

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References

  1. Babson, E., Steingrímsson, E.: Generalized permutation patterns and a classification of the Mahonian statistics. Sém. Lothar. Combin. 44, Art. B44b (2000)

  2. Björner A., Wachs M.: Permutation statistics and linear extensions of posets. J. Combin. Theory Ser. A 58(1), 85–114 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bóna, M.: A combinatorial proof of the log-concavity of a famous sequence counting permutations. Electron. J. Combin. 11(2), #N2 (2005)

  4. Bóna,M.: Combinatorics of Permutations. Chapman & Hall/CRC, Boca Raton, FL (2004)

  5. Brenti, F.: Log-concave and unimodal sequences in algebra, combinatoric, and geometry: an update. In: Barcelo, H., Kalai, G. (eds.) Jerusalem Combinatorics ’93, Contemp. Math. 178, pp. 71–89. Amer. Math. Soc., Providence, RI (1994)

  6. Killpatrick K.: Some statistics for Fibonacci tableaux. European J. Combin. 30(4), 929–933 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. MacMahon, P.A.: Combinatory Analysis. Vol. 1, Cambridge Univ. Press, London (1915)

  8. Rodriguez, O.: Note sur les inversions ou dérangements produits dans les permutations. J. de Mathématique 4, 236–240 (1839)

    Google Scholar 

  9. Stanley R.: The Fibonacci lattice. Fibonacci Quart. 13(3), 215–232 (1975)

    MathSciNet  MATH  Google Scholar 

  10. Stanley, R.: Log-concave and unimodal sequences in algebra, combinatorics, and geometry. In: Capobianco, M.F. et al. (eds.) Graph Theory and Its Applications: East and West, pp. 500–535. New York Acad. Sci., New York (1989)

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Authors and Affiliations

  1. Lewis & Clark College, 0615 S.W. Palatine Hill Road, MSC 47, Portland, OR, 97219, USA

    Naiomi Cameron

  2. Natural Science Division, Pepperdine University, Malibu, California, CA, 90265, USA

    Kendra Killpatrick

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  1. Naiomi Cameron
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  2. Kendra Killpatrick
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Correspondence to Naiomi Cameron.

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Cameron, N., Killpatrick, K. Symmetry and Log-Concavity Results for Statistics on Fibonacci Tableaux. Ann. Comb. 17, 603–618 (2013). https://doi.org/10.1007/s00026-013-0198-1

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