Integral representation of certain combinatorial recurrences

Abstract

Many recurrences that occur in combinatorics incorporate linear and self-convolutive terms. The generating function associated to these is usually not well defined because it has zero radius of convergence. However, the sequence may be identifiable as the asymptotic expansion of a function, and then contour integration can be applied to obtain an expression as the moment sequence of a (possibly signed) measure. We find examples that in combinatorics are all connected with permutations, and whose generating functions are related to the exponential integral function.

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References

  1. [1]

    M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, Dover, New York, 1964.

    Google Scholar 

  2. [2]

    N. Bergeron, C. Hohlweg and M. Zabrocki: Posets related to the connectivity set of Coxeter groups. J. of Algebra 303 (2006), 831–846.

    MathSciNet  Article  Google Scholar 

  3. [3]

    L. Comtet: Sur les coefficients de l’inverse de la série formelle ∑n n!t n, Comptes Rendus Acad. Sci. Paris A275 (1972), 569–572.

    MathSciNet  Google Scholar 

  4. [4]

    L. Comtet: Advanced Combinatorics, Reidel, 1974.

    Google Scholar 

  5. [5]

    P. Flajolet and R. Sedgewick: Analytic Combinatorics, CUP, 2009.

    Google Scholar 

  6. [6]

    R. J. Martin and M. J. Kearney: An exactly solvable self-convolutive recurrence. Aequationes Math. 80 (2010), 291–318.

    MathSciNet  Article  Google Scholar 

  7. [7]

    OEIS Foundation: Online Encyclopedia of Integer Sequences, www.oeis.org.

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Correspondence to Richard J. Martin.

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Martin, R.J., Kearney, M.J. Integral representation of certain combinatorial recurrences. Combinatorica 35, 309–315 (2015). https://doi.org/10.1007/s00493-014-3183-3

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Mathematics Subject Classification (2000)

  • 05A15
  • 30E20
  • 11B37