On Two-Sided Gamma-Positivity for Simple Permutations

Abstract

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 nonnegative 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.

Upper Triangular Linear Relations on Mmultiplicities and the Stanley-Stembridge Conjecture

“…The main result of this section, Theorem 18, gives a set of linear equations satisfied by the multiplicity coefficients c µ and c µ,i of equation (3). In Corollary 23 below, we also reformulate our main result into a family of matrix equations by applying Theorem 18 to the special cases when the set J below is chosen to be J λ for a partition λ of n.…”

“…Theorem 4. Let c µ and c µ,i be the coefficients appearing in (3). Then c µ = c µ,i = 0 for all µ n with more than m(Γ h ) = ht(I h ) + 1 parts.…”

“…Let A = (A(λ, µ)) λ,µ∈Par(n) be the matrix whose coefficients are the integers (14) and let i ∈ Z, i 0. Let X i be the (column) vector whose entries are the c µ,i ∈ Z specified in (3). Let W i be the (column) vector whose entries are the integers |W i (J λ , h)|.…”

“…Example 31. Let n = 6 and λ = (3,3). In this case [5]\J λ = {1, 3, 5}, so if w ∈ D(J λ , J µ ) then (in the one-line notation of w) 2 appears to the left of 1, 4 is to the left of 3, and 6 is to the left of 5.…”

“…We continue Example 52 from above. Here k = 3 and w T = [3,4,2,5,6,1,7,8]. In particular, we have T = {1, 3, 6} and…”