On Two-Sided Gamma-Positivity for Simple Permutations

  • Ron M. Adin Department of Mathematics, Bar-Ilan University,
  • Eli Bagno Jerusalem College of Technology
  • Estrella Eisenberg Jerusalem College of Technology
  • Shulamit Reches Jerusalem College of Technology
  • Moriah Sigron Jerusalem College of Technology
Keywords: Eulerian Polynomials, Gamma positivity, Valley hopping


Gessel conjectured that the two-sided Eulerian polynomial, recording the common distribution of the descent number of a permutation and that of its inverse, has non-negative integer coefficients when expanded in terms of the gamma basis. This conjecture has been proved recently by Lin.

We conjecture that an analogous statement holds for simple permutations, and use the substitution decomposition tree of a permutation (by repeated inflation) to show that this would imply the Gessel-Lin result. We provide supporting evidence for this stronger conjecture.

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