Eulerian quasisymmetric functions

By John Shareshian, Michelle L. Wachs. 2008

101 citations recorded by the source corpus.

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize to enu- merators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric func- tions that Gessel and Reutenauer used to study the joint distribu- tion of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating func- tion for the Eulerian quasisymmetric functions, which specializes to a new and surprising q-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This q-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain q-analogs, (q,p)-analogs and quasisymmetric func- tion analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various represen- tation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Bjorner and Welker.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4fb99b36bbcc5e57a1d9fcd95879d34cc66bcf73

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In Algebra and Number Theory

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Citations in the corpus

Surveys in Combinatorics 2011: The cyclic sieving phenomenon: a survey (2010, background): “one could take one Mahonian and one Eulerian. Given two statistics, st and st′ on a set X we let fst,st′(X;q,t) = X y∈X qstytst′y. Through their study of Rees products of posets, Shareshian and Wachs [72, 73, 74] were lead to consider the pair (maj,exc). They proved, among other things, the followin”
Gamma-positivity in combinatorics and geometry (2017, background): “16 which derive these statements from the main result on Eulerian quasisymmetric functions [105] of Shareshian and Wachs, following the reasoning of [108, Section 4].”
Unimodality, log-concavity, real-rootedness and beyond (2015, background): “n the polynomial X ˇ2Sn S(ˇ)2A xdes(ˇ) is {nonnegative. Postnikov, Reiner and Williams [81] modied the Zn 2-action to prove Gal’s conjecture (see Conjecture 3.6) for so called chordal nestohedra. In [88], Shareshian and Wachs proved renements of the -positivity of Eulerian polynomials. Let A n(q;p;s”
Hopf Algebras in Combinatorics (2014, background): “For variants and generalizations of the Gessel-Reutenauer bijection, see [111], [202], [193], [54] and [172].”
Chromatic quasisymmetric functions (2014, methodology): “tion B.3 that P(q,t) ∈ Q[q][t] is q-positive, q-unimodal and palindromic with center of symmetry 5 2 . 52 SHARESHIANANDWACHS Remark B.4. We have changed the terminology from our previous papers (e.g. [ShWa2]) where we referred to q-positivity and q-unimodality of polynomials in Q[q][t] as t-positivi”