On two-sided gamma-positivity for simple permutations (Q1640211)

Summary: 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 \textit{Z. Lin} [Electron. J. Comb. 23, No. 3, Research Paper P3.15, 9 p. (2016; Zbl 1344.05007)]. 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 [loc. cit.]. We provide supporting evidence for this stronger conjecture.