On Two-Sided Gamma-Positivity for Simple Permutations

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.

Decomposing Sets of Inversions

In this paper we consider the question how the set of inversions of a permutation $\pi \in S_n$ can be partitioned into two subsets, which are themselves inversion sets of permutations in $S_n$. Our method is to study the modular decomposition of the inversion graph of $\pi$. A correspondence to the substitution decomposition of $\pi$ is also given. Moreover, we consider the special case of multiplicative decompositions.

Enumeration of Pin-Permutations

In this paper, we study the class of pin-permutations, that is to say of permutations having a pin representation. This class has been recently introduced by Brignall, Huczynska and Vatter who used it to find properties (algebraicity of the generating function, decidability of membership) of classes of permutations, depending on the simple permutations this class contains. We give a recursive characterization of the substitution decomposition trees of pin-permutations, which allows us to compute the generating function of this class, and consequently to prove, as it is conjectured by Brignall, Ruškuc and Vatter, the rationality of this generating function. Moreover, we show that the basis of the pin-permutation class is infinite.