Gamma-positivity in combinatorics and geometry

Abstract

Gamma-positivity is an elementary property that polynomials with symmetric coefficients may have, which directly implies their unimodality. The idea behind it stems from work of Foata, Schützenberger and Strehl on the Eulerian polynomials; it was revived independently by Brändén and Gal in the course of their study of poset Eulerian polynomials and face enumeration of flag simplicial spheres, respectively, and has found numerous applications since then. This paper surveys some of the main results and open problems on gamma-positivity, appearing in various combinatorial or geometric contexts, as well as some of the diverse methods that have been used to prove it.

Final version

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• Labeled binary trees, subarrangements of the Catalan arrangements, and Schur positivity
• The Eulerian distribution on the involutions of the hyperoctahedral group is unimodal
• Recurrence relations for binomial-Eulerian polynomials
• Specializations of colored quasisymmetric functions and Euler-Mahonian identities
• Positivity and divisibility of alternating descent polynomials
• A unified approach to combinatorial triangles: a generalized Eulerian polynomial
• On the real-rootedness of the Eulerian transformation
• Excedance-type polynomials and gamma-positivity
• Partial $\gamma$-Positivity for Quasi-Stirling Permutations of Multisets
• A symmetry on weakly increasing trees and multiset Schett polynomials
• Eulerian pairs and Eulerian recurrence systems
• Refined Wilf-equivalences by Comtet statistics
• Brändén's $(p,q)$-Eulerian polynomials, André permutations and continued fractions
• Eulerian polynomials and excedance statistics
• Eulerian polynomials for multipermutations and signed multipermutations
• On the joint distribution of cyclic valleys and excedances over conjugacy classes of ${\mathfrak{S}}_n$
• $(q,t)$-Catalan numbers: gamma expansions, pattern avoidance and the $(-1)$-phenomenon
• Gamma-positivity and partial gamma-positivity of descent-type polynomials
• Cycles of even-odd drop permutations and continued fractions of Genocchi numbers
• The gamma-positivity of Eulerian polynomials and succession statistics
• Plethystic formulas for permutation enumeration
• Width-k Eulerian polynomials of type A and B and its Gamma-positivity
• On the unimodality of the Taylor expansion coefficients of Jacobian elliptic functions
• Signed Countings of types B and D permutations and $t,q$-Euler Numbers
• Antichain generating polynomials of posets
• Rooted quasi-Stirling permutations of general multisets
• On the $\mathrm{rix}$ statistic and valley-hopping
• Permanent identities, combinatorial sequences, and permutation statistics
• Koszul Gorenstein algebras from Cohen-Macaulay simplicial complexes
• Eulerian polynomials, Stirling permutations and increasing trees
• The 1/k-Eulerian polynomials of type B
• Refined Catalan and Narayana cyclic sieving
• The alternating run polynomials of permutations
• Around the $q$-binomial-Eulerian polynomials